MathsGee Homework Help Q&A - Recent questions tagged inner
https://mathsgee.com/tag/inner
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 +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 +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 +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 +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 \(w(x)\) 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 +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 \(W\) 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 \(\mathcal{P}_{3}\) be the space of polynomials of degree at most 3 anD let \(D: \mathcal{P}_{3} \rightarrow \mathcal{P}_{3}\) be the derivative operator.
https://mathsgee.com/36361/polynomials-mathcal-rightarrow-mathcal-derivative-operator
Let \(\mathcal{P}_{3}\) be the space of polynomials of degree at most 3 anD let \(D: \mathcal{P}_{3} \rightarrow \mathcal{P}_{3}\) be the derivative operator.<br />
a) Using the basis \(e_{1}=1, e_{2}=x, e_{3}=x^{2}, \epsilon_{4}=x^{3}\) find the matrix \(D_{e}\) representing D.<br />
b) Using the basis \(\epsilon_{1}=x^{3}, \epsilon_{2}=x^{2}, \epsilon_{3}=x, \epsilon_{4}=1\) find the matrix \(D_{e}\) representing D.<br />
c) Show that the matrices \(D_{e}\) and \(D_{e}\) are similar by finding an invertible map \(S\) : \(\mathcal{P}_{3} \rightarrow \mathcal{P}_{3}\) with the property that \(D_{e}=S D_{e} S^{-1}\).Mathematicshttps://mathsgee.com/36361/polynomials-mathcal-rightarrow-mathcal-derivative-operatorFri, 21 Jan 2022 01:21:57 +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 +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 +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 +0000How do ISBN numbers apply dot products?
https://mathsgee.com/35860/how-do-isbn-numbers-apply-dot-products
<p>How do ISBN numbers apply dot products?</p>
<p> </p>
<p><img alt="dot product" src="https://mathsgee.com/?qa=blob&qa_blobid=5890046334045470884" style="height:130px; width:284px"></p>Mathematicshttps://mathsgee.com/35860/how-do-isbn-numbers-apply-dot-productsThu, 13 Jan 2022 09:19:03 +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 +0000If $\mathbf{u}$ and $\mathbf{v}$ are orthogonal vectors in $R^{n}$ with the Euclidean inner product, then prove that $$ \|\mathbf{u}+\mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2} $$
https://mathsgee.com/35820/mathbf-orthogonal-vectors-euclidean-product-mathbf-mathbf
If $\mathbf{u}$ and $\mathbf{v}$ are orthogonal vectors in $R^{n}$ with the Euclidean inner product, then prove that $$ \|\mathbf{u}+\mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2} $$Mathematicshttps://mathsgee.com/35820/mathbf-orthogonal-vectors-euclidean-product-mathbf-mathbfThu, 13 Jan 2022 08:38:00 +0000Let $\mathbf{u}$ and $\mathbf{v}$ be nonzero vectors in 2 - or 3 -space, and let $k=\|\mathbf{u}\|$ and $l=\|\mathbf{v}\|$. Prove that the vector $\mathbf{w}=l \mathbf{u}+k \mathbf{v}$ bisects the angle between $\mathbf{u}$ and $\mathbf{v}$.
https://mathsgee.com/35803/nonzero-vectors-mathbf-mathbf-mathbf-bisects-between-mathbf
Let $\mathbf{u}$ and $\mathbf{v}$ be nonzero vectors in 2 - or 3 -space, and let $k=\|\mathbf{u}\|$ and $l=\|\mathbf{v}\|$.<br />
<br />
Prove that the vector $\mathbf{w}=l \mathbf{u}+k \mathbf{v}$ bisects the angle between $\mathbf{u}$ and $\mathbf{v}$.Mathematicshttps://mathsgee.com/35803/nonzero-vectors-mathbf-mathbf-mathbf-bisects-between-mathbfThu, 13 Jan 2022 08:22:31 +0000What are the basic properties of the inner(dot) product?
https://mathsgee.com/31440/what-are-the-basic-properties-of-the-inner-dot-product
What are the basic properties of the inner(dot) product?Mathematicshttps://mathsgee.com/31440/what-are-the-basic-properties-of-the-inner-dot-productMon, 09 Aug 2021 00:02:53 +0000What is a dot (inner) product?
https://mathsgee.com/31438/what-is-a-dot-inner-product
What is a dot (inner) product?Mathematicshttps://mathsgee.com/31438/what-is-a-dot-inner-productMon, 09 Aug 2021 00:01:26 +0000Show that $\langle v|A| w\rangle=\sum_{i j} A_{i j} v_{i}^{*} w_{j}$
https://mathsgee.com/30911/show-that-langle-v-a-w-rangle-sum-i-j-a-i-j-v-i-w-j
Show that $\langle v|A| w\rangle=\sum_{i j} A_{i j} v_{i}^{*} w_{j}$Mathematicshttps://mathsgee.com/30911/show-that-langle-v-a-w-rangle-sum-i-j-a-i-j-v-i-w-jSat, 24 Jul 2021 09:04:42 +0000Does the Pythagorean theorem generalize to arbitrary inner product spaces?
https://mathsgee.com/27518/pythagorean-theorem-generalize-arbitrary-product-spaces
Does the Pythagorean theorem generalize to arbitrary inner product spaces?Mathematicshttps://mathsgee.com/27518/pythagorean-theorem-generalize-arbitrary-product-spacesSat, 08 May 2021 05:10:12 +0000An inner product on a real vector space $V$ is a function $\langle\cdot, \cdot\rangle: V \times V \rightarrow \mathbb{R}$ satisfying
https://mathsgee.com/27516/product-vector-function-rangle-rightarrow-mathbb-satisfying
An inner product on a real vector space $V$ is a function $\langle\cdot, \cdot\rangle: V \times V \rightarrow \mathbb{R}$ satisfyingMathematicshttps://mathsgee.com/27516/product-vector-function-rangle-rightarrow-mathbb-satisfyingSat, 08 May 2021 05:07:48 +0000Tell me the difference between an inner join, left join/right join, and union.
https://mathsgee.com/16916/tell-difference-between-inner-join-left-join-right-join-union
Tell me the difference between an inner join, left join/right join, and union.Data Science & Statisticshttps://mathsgee.com/16916/tell-difference-between-inner-join-left-join-right-join-unionMon, 27 Jul 2020 20:36:17 +0000