MathsGee Homework Help Q&A - Recent questions tagged product
https://mathsgee.com/tag/product
Powered by Question2AnswerWhat is the dot (Euclidean inner) product of two vectors?
https://mathsgee.com/36637/what-is-the-dot-euclidean-inner-product-of-two-vectors
What is the dot (Euclidean inner) product of two vectors?Mathematicshttps://mathsgee.com/36637/what-is-the-dot-euclidean-inner-product-of-two-vectorsTue, 25 Jan 2022 23:49:30 +0000Knowing the dot product of two vectors, how can I establish the angle between them?
https://mathsgee.com/36635/knowing-the-dot-product-vectors-establish-angle-between-them
Knowing the dot product of two vectors, how can I establish the angle between them?Mathematicshttps://mathsgee.com/36635/knowing-the-dot-product-vectors-establish-angle-between-themTue, 25 Jan 2022 23:48:14 +0000Find the dot product of the vectors shown in the diagram below
https://mathsgee.com/36633/find-the-dot-product-of-the-vectors-shown-in-the-diagram-below
<p>Find the dot product of the vectors shown in the diagram below</p>
<p> </p>
<p><img alt="graph" src="https://mathsgee.com/?qa=blob&qa_blobid=1925024767083500666" style="height:202px; width:259px"></p>Mathematicshttps://mathsgee.com/36633/find-the-dot-product-of-the-vectors-shown-in-the-diagram-belowTue, 25 Jan 2022 23:46:18 +0000What is the component form of the dot product?
https://mathsgee.com/36631/what-is-the-component-form-of-the-dot-product
What is the component form of the dot product?Mathematicshttps://mathsgee.com/36631/what-is-the-component-form-of-the-dot-productTue, 25 Jan 2022 23:44:33 +0000Which mathematician came up with the dot product notation?
https://mathsgee.com/36629/which-mathematician-came-up-with-the-dot-product-notation
Which mathematician came up with the dot product notation?Mathematicshttps://mathsgee.com/36629/which-mathematician-came-up-with-the-dot-product-notationTue, 25 Jan 2022 23:43:21 +0000How do I calculate the dot product of two vectors?
https://mathsgee.com/36627/how-do-i-calculate-the-dot-product-of-two-vectors
How do I calculate the dot product of two vectors?Mathematicshttps://mathsgee.com/36627/how-do-i-calculate-the-dot-product-of-two-vectorsTue, 25 Jan 2022 23:42:12 +0000Calculate $\mathbf{u} \cdot \mathbf{v}$ for the following vectors in $R^{4}$ : $$ \mathbf{u}=(-1,3,5,7), \quad \mathbf{v}=(-3,-4,1,0) $$
https://mathsgee.com/36625/calculate-mathbf-cdot-mathbf-following-vectors-mathbf-mathbf
Calculate $\mathbf{u} \cdot \mathbf{v}$ for the following vectors in $R^{4}$ : $$ \mathbf{u}=(-1,3,5,7), \quad \mathbf{v}=(-3,-4,1,0) $$Mathematicshttps://mathsgee.com/36625/calculate-mathbf-cdot-mathbf-following-vectors-mathbf-mathbfTue, 25 Jan 2022 23:40:54 +0000Find the angle between a diagonal of a cube and one of its edges
https://mathsgee.com/36623/find-the-angle-between-a-diagonal-of-cube-and-one-of-its-edges
Find the angle between a diagonal of a cube and one of its edgesMathematicshttps://mathsgee.com/36623/find-the-angle-between-a-diagonal-of-cube-and-one-of-its-edgesTue, 25 Jan 2022 23:39:31 +0000What are the algebraic properties of the dot/inner product?
https://mathsgee.com/36621/what-are-the-algebraic-properties-of-the-dot-inner-product
What are the algebraic properties of the dot/inner product?Mathematicshttps://mathsgee.com/36621/what-are-the-algebraic-properties-of-the-dot-inner-productTue, 25 Jan 2022 23:38:21 +0000Prove that if $\mathbf{u}$ and $\mathbf{v}$ are vectors in $R^{n}$ with the Euclidean inner product, then $$ \mathbf{u} \cdot \mathbf{v}=\frac{1}{4}\|\mathbf{u}+\mathbf{v}\|^{2}-\frac{1}{4}\|\mathbf{u}-\mathbf{v}\|^{2} $$
https://mathsgee.com/36607/mathbf-vectors-euclidean-product-mathbf-mathbf-mathbf-mathbf
Prove that if $\mathbf{u}$ and $\mathbf{v}$ are vectors in $R^{n}$ with the Euclidean inner product, then $$ \mathbf{u} \cdot \mathbf{v}=\frac{1}{4}\|\mathbf{u}+\mathbf{v}\|^{2}-\frac{1}{4}\|\mathbf{u}-\mathbf{v}\|^{2} $$Mathematicshttps://mathsgee.com/36607/mathbf-vectors-euclidean-product-mathbf-mathbf-mathbf-mathbfTue, 25 Jan 2022 23:28:13 +0000If the product of two numbers is 1050 and their \(\mathrm{HCF}\) is 25 , find their LCM.
https://mathsgee.com/36591/the-product-numbers-1050-and-their-mathrm-hcf-find-their-lcm
If the product of two numbers is 1050 and their \(\mathrm{HCF}\) is 25 , find their LCM.Mathematicshttps://mathsgee.com/36591/the-product-numbers-1050-and-their-mathrm-hcf-find-their-lcmMon, 24 Jan 2022 08:14:46 +0000List the prime factors of 60
https://mathsgee.com/36575/list-the-prime-factors-of-60
List the prime factors of 60Mathematicshttps://mathsgee.com/36575/list-the-prime-factors-of-60Mon, 24 Jan 2022 07:26:21 +0000Let \(v_{1} \ldots v_{k}\) be vectors in a linear space with an inner product \(\langle,\),\(rangle . Define the\) Gram determinant by \(G\left(v_{1}, \ldots, v_{k}\right)=\operatorname{det}\left(\left\langle v_{i}, v_{j}\right\rangle\right)\).
https://mathsgee.com/36470/vectors-product-define-determinant-operatorname-langle-rangle
Let \(v_{1} \ldots v_{k}\) be vectors in a linear space with an inner product \(\langle,\),\(rangle . Define the\) Gram determinant by \(G\left(v_{1}, \ldots, v_{k}\right)=\operatorname{det}\left(\left\langle v_{i}, v_{j}\right\rangle\right)\).<br />
a) If the \(v_{1} \ldots v_{k}\) are orthogonal, compute their Gram determinant.<br />
b) Show that the \(v_{1} \ldots v_{k}\) are linearly independent if and only if their Gram determinant is not zero.<br />
c) Better yet, if the \(v_{1} \ldots v_{k}\) are linearly independent, show that the symmetric matrix \(\left(\left\langle v_{i}, v_{j}\right\rangle\right)\) is positive definite. In particular, the inequality \(G\left(v_{1}, v_{2}\right) \geq 0\) is the Schwarz inequality.<br />
d) Conversely, if \(A\) is any \(n \times n\) positive definite matrix, show that there are vectors \(v_{1}, \ldots, v_{n}\) so that \(A=\left(\left\langle v_{i}, v_{j}\right\rangle\right)\).<br />
e) Let \(\mathcal{S}\) denote the subspace spanned by the linearly independent vectors \(w_{1} \ldots w_{k} .\) If \(X\) is any vector, let \(P_{\mathcal{S}} X\) be the orthogonal projection of \(X\) into \(\mathcal{S}\), prove that the distance \(\left\|X-P_{\mathcal{S}} X\right\|\) from \(X\) to \(\mathcal{S}\) is given by the formula<br />
\[<br />
\left\|X-Z_{\mathcal{S}} X\right\|^{2}=\frac{G\left(X, w_{1}, \ldots, w_{k}\right)}{G\left(w_{1}, \ldots, w_{k}\right)} .<br />
\]Mathematicshttps://mathsgee.com/36470/vectors-product-define-determinant-operatorname-langle-rangleFri, 21 Jan 2022 08:46:45 +0000Let \(A\) be a positive definite \(n \times n\) real matrix, \(\vec{b}\) a real vector, and \(\vec{N}\) a real unit vector.
https://mathsgee.com/36467/positive-definite-times-matrix-real-vector-real-unit-vector
Let \(A\) be a positive definite \(n \times n\) real matrix, \(\vec{b}\) a real vector, and \(\vec{N}\) a real unit vector.<br />
a) For which value(s) of the real scalar \(c\) is the set<br />
\[<br />
E:=\left\{\vec{x} \in \mathbb{R}^{3} \mid\langle\vec{x}, A \vec{x}\rangle+2\langle\vec{b}, \vec{x}\rangle+c=0\right\}<br />
\]<br />
(an ellipsoid) non-empty? <br />
<br />
b) For what value(s) of the scalar \(d\) is the plane \(Z:=\left\{\vec{x} \in \mathbb{R}^{3} \mid\langle\vec{N}, \vec{x}\rangle=d\right\}\) tangent to the above ellipsoid \(E\) (assumed non-empty)?<br />
<br />
[SUGGESTION: First discuss the case where \(A=I\) and \(\vec{b}=0\). Then show how by a change of variables, the general case can be reduced to this special case. ]Mathematicshttps://mathsgee.com/36467/positive-definite-times-matrix-real-vector-real-unit-vectorFri, 21 Jan 2022 08:41:48 +0000Let \(\vec{x}\) and \(\vec{p}\) be points in \(\mathbb{R}^{n}\). Under what conditions on the scalar \(c\) is the set \[ \|\vec{x}\|^{2}+2\langle\vec{p}, \vec{x}\rangle+c=0 \]
https://mathsgee.com/36466/points-mathbb-under-what-conditions-the-scalar-langle-rangle
a) Let \(\vec{x}\) and \(\vec{p}\) be points in \(\mathbb{R}^{n}\). Under what conditions on the scalar \(c\) is the set<br />
\[<br />
\|\vec{x}\|^{2}+2\langle\vec{p}, \vec{x}\rangle+c=0<br />
\]<br />
a sphere \(\left\|\vec{x}-\vec{x}_{0}\right\|=R \geq 0\) ? Compute the center, \(\vec{x}_{0}\), and radius, \(R\), in terms of \(\vec{p}\) and \(c\).<br />
b) Let<br />
\[<br />
\begin{aligned}<br />
Q(\vec{x}) &=\sum a_{i j} x_{i} x_{j}+2 \sum b_{i} x_{i}+c \\<br />
&=\langle\vec{x}, A \vec{x}\rangle+2\langle\vec{b}, \vec{x}\rangle+c<br />
\end{aligned}<br />
\]<br />
be a real quadratic polynomial so \(\vec{x}=\left(x_{1}, \ldots, x_{n}\right), \vec{b}=\left(b_{1}, \ldots, b_{n}\right)\) are real vectors and \(A=\left(a_{i j}\right)\) is a real symmetric \(n \times n\) matrix. Just as in the case \(n=1\) (which you should do first), if \(A\) is invertible find a vector \(\vec{v}\) (depending on \(A\) and \(\vec{b}\) ) so that the change of variables \(\vec{y}==\vec{x}-\vec{v}\) (this is a translation by the vector \(\vec{v}\) ) so that in the new \(\vec{y}\) variables \(Q\) has the simpler form<br />
\[<br />
Q=\langle\vec{y}, A \vec{y}\rangle+\gamma \text { that is, } Q=\sum a_{i j} y_{i} y_{j}+\gamma,<br />
\]<br />
where \(\gamma=c-\left\langle\vec{b}, A^{-1} \vec{b}\right\rangle\).<br />
As an example, apply this to \(Q(\vec{x})=2 x_{1}^{2}+2 x_{1} x_{2}+3 x_{2}-4\).Mathematicshttps://mathsgee.com/36466/points-mathbb-under-what-conditions-the-scalar-langle-rangleFri, 21 Jan 2022 08:39:47 +0000Find the function \(f \in \operatorname{span}\{1 \sin x, \cos x\}\) that minimizes \(\|\sin 2 x-f(x)\|\), where the norm comes from the inner product
https://mathsgee.com/36463/function-operatorname-minimizes-where-comes-inner-product
Find the function \(f \in \operatorname{span}\{1 \sin x, \cos x\}\) that minimizes \(\|\sin 2 x-f(x)\|\), where the norm comes from the inner product<br />
\[<br />
\langle f, g\rangle:=\int_{-\pi}^{\pi} f(x) g(x) d x \quad \text { on } \quad C[-\pi, \pi] .<br />
\]Mathematicshttps://mathsgee.com/36463/function-operatorname-minimizes-where-comes-inner-productFri, 21 Jan 2022 08:35:56 +0000Let \(C[-1,1]\) be the real inner product space consisting of all continuous functions \(f:[-1,1] \rightarrow \mathbb{R}\), with the inner product \(\langle f, g\rangle:=\int_{-1}^{1} f(x) g(x) d x\).
https://mathsgee.com/36462/product-consisting-continuous-functions-rightarrow-product
Let \(C[-1,1]\) be the real inner product space consisting of all continuous functions \(f:[-1,1] \rightarrow \mathbb{R}\), with the inner product \(\langle f, g\rangle:=\int_{-1}^{1} f(x) g(x) d x\). Let \(W\) be the subspace of odd functions, i.e. functions satisfying \(f(-x)=-f(x)\). Find (with proof) the orthogonal complement of \(W\).Mathematicshttps://mathsgee.com/36462/product-consisting-continuous-functions-rightarrow-productFri, 21 Jan 2022 08:34:34 +0000Let \(\mathcal{P}_{2}\) be the space of polynomials \(p(x)=a+b x+c x^{2}\) of degree at most 2 with the inner product \(\langle p, q\rangle=\int_{-1}^{1} p(x) q(x) d x\).
https://mathsgee.com/36461/mathcal-space-polynomials-degree-inner-product-langle-rangle
Let \(\mathcal{P}_{2}\) be the space of polynomials \(p(x)=a+b x+c x^{2}\) of degree at most 2 with the inner product \(\langle p, q\rangle=\int_{-1}^{1} p(x) q(x) d x\). Let \(\ell\) be the functional \(\ell(p):=p(0)\). Find \(h \in \mathcal{P}_{2}\) so that \(\ell(p)=\langle h, p\rangle\) for all \(p \in \mathcal{P}_{2}\).Mathematicshttps://mathsgee.com/36461/mathcal-space-polynomials-degree-inner-product-langle-rangleFri, 21 Jan 2022 08:33:56 +0000Let \(\mathcal{P}_{2}\) be the space of quadratic polynomials.
https://mathsgee.com/36460/let-mathcal-p-2-be-the-space-of-quadratic-polynomials
Let \(\mathcal{P}_{2}\) be the space of quadratic polynomials.<br />
<br />
a) Show that \(\langle f, g\rangle=f(-1) g(-1)+f(0) g(0)+f(1) g(1)\) is an inner product for this space.<br />
<br />
b) Using this inner product, find an orthonormal basis for \(\mathcal{P}_{2}\).<br />
<br />
c) Is this also an inner product for the space \(\mathcal{P}_{3}\) of polynomials of degree at most three? Why?Mathematicshttps://mathsgee.com/36460/let-mathcal-p-2-be-the-space-of-quadratic-polynomialsFri, 21 Jan 2022 08:33:18 +0000Using the inner product of the previous problem, let \(\mathcal{B}=\left\{1, x, 3 x^{2}-1\right\}\) be an orthogonal basis for the space \(\mathcal{P}_{2}\) of quadratic polynomials and . . .
https://mathsgee.com/36458/product-previous-problem-orthogonal-quadratic-polynomials
Using the inner product of the previous problem, let \(\mathcal{B}=\left\{1, x, 3 x^{2}-1\right\}\) be an orthogonal basis for the space \(\mathcal{P}_{2}\) of quadratic polynomials and let \(\mathcal{S}=\operatorname{span}\left(x, x^{2}\right) \subset\) \(\mathcal{P}_{2}\). Using the basis \(\mathcal{B}\), find the linear map \(P: \mathcal{P}_{2} \rightarrow \mathcal{P}_{2}\) that is the orthogonal projection from \(\mathcal{P}_{2}\) onto \(\mathcal{S}\).Mathematicshttps://mathsgee.com/36458/product-previous-problem-orthogonal-quadratic-polynomialsFri, 21 Jan 2022 08:30:42 +0000Using the inner product \(\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x\), for which values of the real constants \(\alpha, \beta, \gamma\) are the quadratic polynomials ...
https://mathsgee.com/36457/product-langle-rangle-values-constants-quadratic-polynomials
Using the inner product \(\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x\), for which values of the real constants \(\alpha, \beta, \gamma\) are the quadratic polynomials \(p_{1}(x)=1, \quad p_{2}(x)=\alpha+x \quad p_{3}(x)=\beta+\gamma x+x^{2}\) orthogonal? ]Mathematicshttps://mathsgee.com/36457/product-langle-rangle-values-constants-quadratic-polynomialsFri, 21 Jan 2022 08:29:12 +0000In a complex vector space (with a hermitian inner product), if a matrix \(A\) satisfies \(\langle X, A X\rangle=0\) for all vectors \(X\), show that \(A=0\). [The previous problem shows that this is false in a real vector space].
https://mathsgee.com/36456/complex-hermitian-product-satisfies-vectors-previous-problem
In a complex vector space (with a hermitian inner product), if a matrix \(A\) satisfies \(\langle X, A X\rangle=0\) for all vectors \(X\), show that \(A=0\). [The previous problem shows that this is false in a real vector space].Mathematicshttps://mathsgee.com/36456/complex-hermitian-product-satisfies-vectors-previous-problemFri, 21 Jan 2022 08:27:16 +0000Proof or counterexample. Here \(v, w, z\) are vectors in a real inner product space \(H\).
https://mathsgee.com/36451/proof-counterexample-here-vectors-real-inner-product-space
Proof or counterexample. Here \(v, w, z\) are vectors in a real inner product space \(H\).<br />
a) Let \(v, w, z\) be vectors in a real inner product space. If \(\langle v, w\rangle=0\) and \(\langle v, z\rangle=0\), then \(\langle w, z\rangle=0\).<br />
b) If \(\langle v, z\rangle=\langle w, z\rangle\) for all \(z \in H\), then \(v=w .\)<br />
c) If \(A\) is an \(n \times n\) symmetric matrix then \(A\) is invertible.Mathematicshttps://mathsgee.com/36451/proof-counterexample-here-vectors-real-inner-product-spaceFri, 21 Jan 2022 08:19:29 +0000Let \(A\) be a positive definite \(n \times n\) real matrix, \(b \in \mathbb{R}^{n}\), and consider the quadratic polynomial \[ Q(x):=\frac{1}{2}\langle x, A x\rangle-\langle b, x\rangle \]
https://mathsgee.com/36446/positive-definite-mathbb-consider-quadratic-polynomial-langle
Let \(A\) be a positive definite \(n \times n\) real matrix, \(b \in \mathbb{R}^{n}\), and consider the quadratic polynomial<br />
\[<br />
Q(x):=\frac{1}{2}\langle x, A x\rangle-\langle b, x\rangle<br />
\]<br />
a) Show that \(Q\) is bounded below, that is, there is a constant \(m\) so that \(Q(x) \geq m\) for all \(x \in \mathbb{R}^{n}\).<br />
b) Show that \(Q\) blows up at infinity by showing that there are positive constants \(R\) and \(c\) so that if \(\|x\| \geq R\), then \(Q(x) \geq c\|x\|^{2}\).<br />
33<br />
c) If \(x_{0} \in \mathbb{R}^{n}\) minimizes \(Q\), show that \(A x_{0}=b\). [Moral: One way to solve \(A x=b\) is to minimize \(Q .\) ]<br />
d) Give an example showing that if \(A\) is only positive semi-definite, then \(Q(x)\) may not be bounded below.Mathematicshttps://mathsgee.com/36446/positive-definite-mathbb-consider-quadratic-polynomial-langleFri, 21 Jan 2022 08:15:07 +0000Let \(\ell\) be any linear functional. Show there is a unique vector \(v \in \mathbb{R}^{n}\) so that \(\ell(x):=\langle x, v\rangle\).
https://mathsgee.com/36445/linear-functional-there-unique-vector-mathbb-langle-rangle
[LINEAR FUNCTIONALS] In \(R^{n}\) with the usual inner product, a linear functional \(\ell:\) \(\mathbb{R}^{n} \rightarrow \mathbb{R}\) is just a linear map into the reals (in a complex vector space, it maps into the complex numbers \(\mathbb{C}\) ). Define the norm, \(\|\ell\|\), as<br />
\[<br />
\|\ell\|:=\max _{\|x\|=1}|\ell(x)| .<br />
\]<br />
a) Show that the set of linear functionals with this norm is a normed linear space.<br />
b) If \(v \in \mathbb{R}^{n}\) is a given vector, define \(\ell(x)=\langle x, v\rangle\). Show that \(\ell\) is a linear functional and that \(\|\ell\|=\|v\|\).<br />
c) [REPRESENTATION OF A LINEAR FUNCTIONAL] Let \(\ell\) be any linear functional. Show there is a unique vector \(v \in \mathbb{R}^{n}\) so that \(\ell(x):=\langle x, v\rangle\).<br />
d) [EXTENSION OF A LINEAR FUNCTIONAL] Let \(U \subset \mathbb{R}^{n}\) be a subspace of \(\mathbb{R}^{n}\) and \(\ell\) a linear functional defined on \(U\) with norm \(\|\ell\|_{U}\). Show there is a unique extension of \(\ell\) to \(\mathbb{R}^{n}\) with the property that \(\|\ell\|_{\mathbb{R}^{n}}=\|\ell\|_{U}\).<br />
[In other words define \(\ell\) on all of \(\mathbb{R}^{n}\) so that on \(U\) this extended definition agrees with the original definition and so that its norm is unchanged].Mathematicshttps://mathsgee.com/36445/linear-functional-there-unique-vector-mathbb-langle-rangleFri, 21 Jan 2022 08:14:22 +0000Let \(w(x)\) be a positive continuous function on the interval \(0 \leq x \leq 1, n\) a positive integer, and \(\mathcal{P}_{n}\) the vector space of polynomials \(p(x)\) whose degrees are at most \(n\) equipped with the inner product
https://mathsgee.com/36444/positive-continuous-function-interval-polynomials-equipped
Let \(w(x)\) be a positive continuous function on the interval \(0 \leq x \leq 1, n\) a positive integer, and \(\mathcal{P}_{n}\) the vector space of polynomials \(p(x)\) whose degrees are at most \(n\) equipped with the inner product<br />
\[<br />
\langle p, q\rangle=\int_{0}^{1} p(x) q(x) w(x) d x .<br />
\]<br />
a) Prove that \(\mathcal{P}_{n}\) has an orthonormal basis \(p_{0}, p_{1}, \ldots, p_{n}\) with the degree of \(p_{k}\) is \(k\) for each \(k\).<br />
b) Prove that \(\left\langle p_{k}, p_{k}^{\prime}\right\rangle=0\) for each \(k\).Mathematicshttps://mathsgee.com/36444/positive-continuous-function-interval-polynomials-equippedFri, 21 Jan 2022 08:12:34 +0000Let \(V\) be the real vector space of continuous real-valued functions on the closed interval \([0,1]\), and let \(w \in V\). For \(p, q \in V\), define \(\langle p, q\rangle=\int_{0}^{1} p(x) q(x) w(x) d x\).
https://mathsgee.com/36443/vector-continuous-functions-interval-define-langle-rangle
Let \(V\) be the real vector space of continuous real-valued functions on the closed interval \([0,1]\), and let \(w \in V\). For \(p, q \in V\), define \(\langle p, q\rangle=\int_{0}^{1} p(x) q(x) w(x) d x\).<br />
a) Suppose that \(w(a)>0\) for all \(a \in[0,1]\). Does it follow that the above defines an inner product on \(V\) ? Justify your assertion.<br />
<br />
b) Does there exist a choice of \(w\) such that \(w(1 / 2)<0\) and such that the above defines an inner product on \(V ?\) Justify your assertion.Mathematicshttps://mathsgee.com/36443/vector-continuous-functions-interval-define-langle-rangleFri, 21 Jan 2022 08:11:57 +0000Let \(U, V, W\) be orthogonal vectors and let \(Z=a U+b V+c W\), where \(a, b, c\) are scalars.
https://mathsgee.com/36441/let-u-be-orthogonal-vectors-and-let-z-a-b-v-c-w-where-are-scalars
Let \(U, V, W\) be orthogonal vectors and let \(Z=a U+b V+c W\), where \(a, b, c\) are scalars.<br />
a) (Pythagoras) Show that \(\|Z\|^{2}=a^{2}\|U\|^{2}+b^{2}\|V\|^{2}+c^{2}\|W\|^{2}\).<br />
b) Find a formula for the coefficient \(a\) in terms of \(U\) and \(Z\) only. Then find similar formulas for \(b\) and \(c\). [Suggestion: take the inner product of \(Z=a U+b V+c W\) with \(U\) ].<br />
REMARK The resulting simple formulas are one reason that orthogonal vectors are easier to use than more general vectors. This is vital for Fourier series.<br />
c) Solve the following equations:<br />
\[<br />
\begin{aligned}<br />
&x+y+z+w=2 \\<br />
&x+y-z-w=3 \\<br />
&x-y+z-w=0 \\<br />
&x-y-z+w=-5<br />
\end{aligned}<br />
\]<br />
[Suggestion: Observe that the columns vectors in the coefficient matrix are orthogonal.]Mathematicshttps://mathsgee.com/36441/let-u-be-orthogonal-vectors-and-let-z-a-b-v-c-w-where-are-scalarsFri, 21 Jan 2022 08:09:48 +0000Find all vectors in the plane (through the origin) spanned by \(\mathbf{V}=(1,1-2)\) and \(\mathbf{W}=(-1,1,1)\) that are perpendicular to the vector \(\mathbf{Z}=(2,1,2)\).
https://mathsgee.com/36438/vectors-through-origin-spanned-perpendicular-vector-mathbf
Find all vectors in the plane (through the origin) spanned by \(\mathbf{V}=(1,1-2)\) and \(\mathbf{W}=(-1,1,1)\) that are perpendicular to the vector \(\mathbf{Z}=(2,1,2)\).Mathematicshttps://mathsgee.com/36438/vectors-through-origin-spanned-perpendicular-vector-mathbfFri, 21 Jan 2022 08:07:00 +0000Let \(V, W\) be vectors in \(\mathbb{R}^{n}\).
https://mathsgee.com/36435/let-v-w-be-vectors-in-mathbb-r-n
Let \(V, W\) be vectors in \(\mathbb{R}^{n}\).<br />
<br />
a) Show that the Pythagorean relation \(\|V+W\|^{2}=\|V\|^{2}+\|W\|^{2}\) holds if and only if \(V\) and \(W\) are orthogonal.<br />
<br />
b) Prove the parallelogram identity \(\|V+W\|^{2}+\|V-W\|^{2}=2\|V\|^{2}+2\|W\|^{2}\) and interpret it geometrically. [This is true in any real inner product space].Mathematicshttps://mathsgee.com/36435/let-v-w-be-vectors-in-mathbb-r-nFri, 21 Jan 2022 08:04:40 +0000Let \(A\) be an \(m \times n\) matrix, and suppose \(\vec{v}\) and \(\vec{w}\) are orthogonal eigenvectors of \(A^{T} A\). Show that \(A \vec{v}\) and \(A \vec{w}\) are orthogonal.
https://mathsgee.com/36408/times-matrix-suppose-orthogonal-eigenvectors-orthogonal
Let \(A\) be an \(m \times n\) matrix, and suppose \(\vec{v}\) and \(\vec{w}\) are orthogonal eigenvectors of \(A^{T} A\). Show that \(A \vec{v}\) and \(A \vec{w}\) are orthogonal.Mathematicshttps://mathsgee.com/36408/times-matrix-suppose-orthogonal-eigenvectors-orthogonalFri, 21 Jan 2022 02:05:03 +0000Let \(W\) be a linear space with an inner product and \(A: W \rightarrow W\) be a linear map whose image is one dimensional (so in the case of matrices, it has rank one).
https://mathsgee.com/36377/linear-product-rightarrow-linear-image-dimensional-matrices
Let \(W\) be a linear space with an inner product and \(A: W \rightarrow W\) be a linear map whose image is one dimensional (so in the case of matrices, it has rank one). Let \(\vec{v} \neq 0\) be in the image of \(A\), so it is a basis for the image. If \(\langle\vec{v},(I+A) \vec{v}\rangle \neq 0\), show that \(I+A\) is invertible by finding a formula for the inverse.Mathematicshttps://mathsgee.com/36377/linear-product-rightarrow-linear-image-dimensional-matricesFri, 21 Jan 2022 01:38:13 +0000Let \(\vec{v}\) and \(\vec{w}\) be vectors in \(\mathbb{R}^{n}\). If \(\|\vec{v}\|=\|\vec{w}\|\), show there is an orthogonal matrix \(R\) with \(R \vec{v}=\vec{w}\) and \(R \vec{w}=\vec{v}\).
https://mathsgee.com/36374/vectors-mathbb-vec-show-there-orthogonal-matrix-with-vec-vec
Let \(\vec{v}\) and \(\vec{w}\) be vectors in \(\mathbb{R}^{n}\). If \(\|\vec{v}\|=\|\vec{w}\|\), show there is an orthogonal matrix \(R\) with \(R \vec{v}=\vec{w}\) and \(R \vec{w}=\vec{v}\).Mathematicshttps://mathsgee.com/36374/vectors-mathbb-vec-show-there-orthogonal-matrix-with-vec-vecFri, 21 Jan 2022 01:34:09 +0000Let \(\mathcal{P}_{2}\) be the space of polynomials of degree at most 2 .
https://mathsgee.com/36360/let-mathcal-p-2-be-the-space-of-polynomials-of-degree-at-most
Let \(\mathcal{P}_{2}\) be the space of polynomials of degree at most 2 .<br />
<br />
<br />
<br />
a) Find a basis for this space.<br />
b) Let \(D: \mathcal{P}_{2} \rightarrow \mathcal{P}_{2}\) be the derivative operator \(D=d / d x\). Using the basis you picked in the previous part, write \(D\) as a matrix. Compute \(D^{3}\) in this situation. Why should you have predicted this without computation?Mathematicshttps://mathsgee.com/36360/let-mathcal-p-2-be-the-space-of-polynomials-of-degree-at-mostFri, 21 Jan 2022 01:20:47 +0000Let \(\mathcal{P}_{1}\) be the linear space of real polynomials of degree at most one, so a typical element is \(p(x):=a+b x\), where \(a\) and \(b\) are real numbers.
https://mathsgee.com/36359/mathcal-linear-polynomials-degree-typical-element-numbers
Let \(\mathcal{P}_{1}\) be the linear space of real polynomials of degree at most one, so a typical element is \(p(x):=a+b x\), where \(a\) and \(b\) are real numbers. The derivative, \(D: \mathcal{P}_{1} \rightarrow \mathcal{P}_{1}\) is, as you should expect, the map \(D P(x)=b=b+0 x\). Using the basis \(e_{1}(x):=1\), \(e_{2}(x):=x\) for \(\mathcal{P}_{1}\), we have \(p(x)=a e_{1}(x)+b e_{2}(x)\) so \(D p=b e_{1}\).<br />
Using this basis, find the \(2 \times 2\) matrix \(M\) for \(D\). Note the obvious property \(D^{2} p=0\) for any polynomial \(p\) of degree at most 1 . Does \(M\) also satisfy \(M^{2}=0\) ? Why should you have expected this?Mathematicshttps://mathsgee.com/36359/mathcal-linear-polynomials-degree-typical-element-numbersFri, 21 Jan 2022 01:20:18 +0000If \(A\) and \(B\) are \(4 \times 4\) matrices such that \(\operatorname{rank}(A B)=3\), then \(\operatorname{rank}(B A)<4\).
https://mathsgee.com/36336/times-matrices-such-operatorname-rank-then-operatorname-rank
For each of the following, answer TRUE or FALSE. If the statement is false in even a single instance, then the answer is FALSE. There is no need to justify your answers to this problem - but you should know either a proof or a counterexample.<br />
<br />
a) If \(A\) and \(B\) are \(4 \times 4\) matrices such that \(\operatorname{rank}(A B)=3\), then \(\operatorname{rank}(B A)<4\).<br />
b) If \(A\) is a \(5 \times 3\) matrix with \(\operatorname{rank}(A)=2\), then for every vector \(b \in \mathbb{R}^{5}\) the equation \(A x=b\) will have at least one solution.<br />
c) If \(A\) is a \(4 \times 7\) matrix, then \(A\) and \(A^{T}\) have the same rank.<br />
d) Let \(A\) and \(B \neq 0\) be \(2 \times 2\) matrices. If \(A B=0\), then \(A\) must be the zero matrix.Mathematicshttps://mathsgee.com/36336/times-matrices-such-operatorname-rank-then-operatorname-rankFri, 21 Jan 2022 01:00:05 +0000Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map. Show that the following are equivalent.
https://mathsgee.com/36310/mathbb-rightarrow-mathbb-linear-that-following-equivalent
Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map. Show that the following are equivalent.<br />
a) For every \(y \in \mathbb{R}^{k}\) the equation \(A x=y\) has at most one solution.<br />
b) \(A\) is injective (hence \(n \leq k\) ). [injective means one-to-one]<br />
c) \(\operatorname{dim} \operatorname{ker}(A)=0\).<br />
d) \(A^{*}\) is surjective (onto).<br />
e) The columns of \(A\) are linearly independent.Mathematicshttps://mathsgee.com/36310/mathbb-rightarrow-mathbb-linear-that-following-equivalentThu, 20 Jan 2022 13:02:54 +0000Let \(\mathcal{P}_{k}\) be the space of polynomials of degree at most \(k\) and define the linear map \(L: \mathcal{P}_{k} \rightarrow \mathcal{P}_{k+1}\) by \(L p:=p^{\prime \prime}(x)+x p(x) .\)
https://mathsgee.com/36301/mathcal-polynomials-define-linear-mathcal-rightarrow-mathcal
Let \(\mathcal{P}_{k}\) be the space of polynomials of degree at most \(k\) and define the linear map \(L: \mathcal{P}_{k} \rightarrow \mathcal{P}_{k+1}\) by \(L p:=p^{\prime \prime}(x)+x p(x) .\)<br />
<br />
<br />
<br />
a) Show that the polynomial \(q(x)=1\) is not in the image of \(L\). [SUGGESTION: Try the case \(k=2\) first.]<br />
b) Let \(V=\left\{q(x) \in \mathcal{P}_{k+1} \mid q(0)=0\right\}\). Show that the map \(L: \mathcal{P}_{k} \rightarrow V\) is invertible. [Again, try \(k=2\) first.]Mathematicshttps://mathsgee.com/36301/mathcal-polynomials-define-linear-mathcal-rightarrow-mathcalThu, 20 Jan 2022 12:48:12 +0000What are the rules of differentiation?
https://mathsgee.com/35923/what-are-the-rules-of-differentiation
What are the rules of differentiation?Mathematicshttps://mathsgee.com/35923/what-are-the-rules-of-differentiationSat, 15 Jan 2022 02:50:24 +0000Verify that the Cauchy-Schwarz inequality holds. \(\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)\)
https://mathsgee.com/35875/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbf
Verify that the Cauchy-Schwarz inequality holds.<br />
<br />
(b) \(\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)\)Mathematicshttps://mathsgee.com/35875/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbfFri, 14 Jan 2022 09:59:16 +0000What is the dot product of $\mathbf{u}$ a column matrix and $\mathbf{v}$ a column matrix?
https://mathsgee.com/35866/what-product-mathbf-column-matrix-and-mathbf-column-matrix
What is the dot product of $\mathbf{u}$ a column matrix and $\mathbf{v}$ a column matrix?Mathematicshttps://mathsgee.com/35866/what-product-mathbf-column-matrix-and-mathbf-column-matrixThu, 13 Jan 2022 09:24:26 +0000What is the dot product of u a row matrix and v a column matrix?
https://mathsgee.com/35864/what-is-the-dot-product-of-u-a-row-matrix-and-v-a-column-matrix
What is the dot product of u a row matrix and v a column matrix?Mathematicshttps://mathsgee.com/35864/what-is-the-dot-product-of-u-a-row-matrix-and-v-a-column-matrixThu, 13 Jan 2022 09:22:34 +0000What is the dot product of u a column matrix and v a row matrix?
https://mathsgee.com/35862/what-is-the-dot-product-of-u-a-column-matrix-and-v-a-row-matrix
What is the dot product of u a column matrix and v a row matrix?Mathematicshttps://mathsgee.com/35862/what-is-the-dot-product-of-u-a-column-matrix-and-v-a-row-matrixThu, 13 Jan 2022 09:21:00 +0000Verify that the Cauchy–Schwarz inequality holds for $\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$
https://mathsgee.com/35854/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbf
Verify that the Cauchy&ndash;Schwarz inequality holds for $\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$Mathematicshttps://mathsgee.com/35854/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbfThu, 13 Jan 2022 09:13:52 +0000Verify that the Cauchy–Schwarz inequality holds for $\mathbf{u}=(4,1,1), \mathbf{v}=(1,2,3)$
https://mathsgee.com/35853/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbf
Verify that the Cauchy&ndash;Schwarz inequality holds for $\mathbf{u}=(4,1,1), \mathbf{v}=(1,2,3)$Mathematicshttps://mathsgee.com/35853/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbfThu, 13 Jan 2022 09:13:03 +0000Verify that the Cauchy–Schwarz inequality holds for $\mathbf{u}=(1,2,1,2,3), \mathbf{v}=(0,1,1,5,-2)$
https://mathsgee.com/35852/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbf
Verify that the Cauchy&ndash;Schwarz inequality holds for $\mathbf{u}=(1,2,1,2,3), \mathbf{v}=(0,1,1,5,-2)$Mathematicshttps://mathsgee.com/35852/verify-that-cauchy-schwarz-inequality-holds-mathbf-mathbfThu, 13 Jan 2022 09:12:11 +0000Let $\mathbf{r}_{0}=\left(x_{0}, y_{0}\right)$ be a fixed vector in $R^{2}$. In each part, describe in words the set of all vectors $\mathbf{r}=(x, y)$ that satisfy the stated condition.
https://mathsgee.com/35851/mathbf-vector-describe-vectors-mathbf-satisfy-stated-condition
Let $\mathbf{r}_{0}=\left(x_{0}, y_{0}\right)$ be a fixed vector in $R^{2}$. In each part, describe in words the set of all vectors $\mathbf{r}=(x, y)$ that satisfy the stated condition. (a) $\left\|\mathbf{r}-\mathbf{r}_{0}\right\|=1$ (b) $\left\|\mathbf{r}-\mathbf{r}_{0}\right\| \leq 1$ (c) $\left\|\mathbf{r}-\mathbf{r}_{0}\right\|>1$Mathematicshttps://mathsgee.com/35851/mathbf-vector-describe-vectors-mathbf-satisfy-stated-conditionThu, 13 Jan 2022 09:11:15 +0000Show that two nonzero vectors $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ in $R^{3}$ are orthogonal if and only if their direction cosines satisfy
https://mathsgee.com/35848/nonzero-vectors-mathbf-orthogonal-direction-cosines-satisfy
Show that two nonzero vectors $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ in $R^{3}$ are orthogonal if and only if their direction cosines satisfy $$ \cos \alpha_{1} \cos \alpha_{2}+\cos \beta_{1} \cos \beta_{2}+\cos \gamma_{1} \cos \gamma_{2}=0 $$Mathematicshttps://mathsgee.com/35848/nonzero-vectors-mathbf-orthogonal-direction-cosines-satisfyThu, 13 Jan 2022 09:08:14 +0000Let $\mathbf{u}$ be a vector in $R^{100}$ whose $i$ th component is $i$, and let $\mathbf{v}$ be the vector in $R^{100}$ whose $i$ th component is $1 /(i+1)$. Find the dot product of $\mathbf{u}$ and $\mathbf{v}$.
https://mathsgee.com/35845/mathbf-vector-component-vector-component-product-mathbf-mathbf
Let $\mathbf{u}$ be a vector in $R^{100}$ whose $i$ th component is $i$, and let $\mathbf{v}$ be the vector in $R^{100}$ whose $i$ th component is $1 /(i+1)$. Find the dot product of $\mathbf{u}$ and $\mathbf{v}$.Mathematicshttps://mathsgee.com/35845/mathbf-vector-component-vector-component-product-mathbf-mathbfThu, 13 Jan 2022 09:04:24 +0000When are two nonzero vectors orthogonal to each other?
https://mathsgee.com/35842/when-are-two-nonzero-vectors-orthogonal-to-each-other
When are two nonzero vectors orthogonal to each other?Mathematicshttps://mathsgee.com/35842/when-are-two-nonzero-vectors-orthogonal-to-each-otherThu, 13 Jan 2022 09:00:47 +0000