Recent questions tagged inner

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Let $L = D - W \in \mathbb{R}^{n \times n}$ be the graph Laplacian for data with an associated symmetric weight matrix $W$, and $w_{ij} \in [0,1]$ for all $i,j = 1,...,n$...
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(a) Prove that $\langle x,y\rangle_M = x M y^T$ satisfies the properties of an inner product if $M$ is positive definite.(b) Show that $\langle x,y\rangle_M$ need not be ...
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Prove that the Euclidean dot product $\langle x, y \rangle = \sum_{i=1}^n x_i y_i, x, y \in \mathbb{R}^n$ is an inner product, where an inner product is a binary function...
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A vector in standard form has components \(\langle 3,10\rangle\). What is the initial point?
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Was the Pythagorean School a cult?
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What was the significance of the invention of logarithms?
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What was the significance of the Pythagorean Theorem?
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What is the dot product of two vectors?
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What is the Pythagorean Theorem?
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What is the Pythagorean theorem?
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Multiply the polynomials $(x+2)(x+3)$ and $(x-1)(x-4)$.
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Calculate the product of \(2 x-1\) and \(x^2+2 x-3\).
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Find all functions \(f: \mathbb{N} \rightarrow \mathbb{N}\) such that (i) if \(x<y\), then \(f(x)<f(y)\)(ii) \(f(y f(x))=x^2 f(x y)\) for all \(x, y \in \mathbb{N}\).
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Find all functions \(f\) such that\[f(f(x)+y)=f\left(x^2-y\right)+4 y f(x)\]for all real numbers \(x\) and \(y\).
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Find all functions \(f\) from \(\mathbb{R} \backslash\{0\}\) to itself satisfying\[f(x)+f(y)=f(x y f(x+y))\]for all \(x, y \neq 0\) with \(x+y \neq 0\).
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Find all functions \(f: \mathbb{Q} \rightarrow \mathbb{Q}\) such that\[f(x+y)+f(x-y)=2 f(x)+2 f(y) \quad \text { for all } x, y \in \mathbb{Q}\]
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Determine all functions \(f: \mathbb{Q} \rightarrow \mathbb{Q}\) such that\[f(x f(y))=(1-y) f(x y)+x^2 y^2 f(y)\]holds for all real numbers \(x\) and \(y\).
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Find all functions \(f: \mathbb{N}_0 \rightarrow \mathbb{N}_0\) whose minimum value is 1 and \(f(f(n))=f(n)+1 \quad\) for all \(n \in \mathbb{N}_0\)
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Find all functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that for all \(x, y\)\[f\left(x^2\right)-f\left(y^2\right)=(x+y)(f(x)-f(y)) .\]
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Does there exist a function \(f: \mathbb{N}_0 \rightarrow \mathbb{N}_0\) such that\[\underbrace{f(f(\ldots f}_{2003}(n) \ldots))=5 n \quad \text { for all } n \in \mathbb...
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Find all functions \(f: \mathbb{N}_0 \rightarrow \mathbb{N}_0\) satisfying \(f(2)=2, f(n)<f(n+1)\) and \(f(m n)=\) \(f(m) f(n)\) for all \(m, n \in \mathbb{N}\)
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How do I round real numbers to an appropriate degree of accuracy?
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Let \(\mathrm{V}\) be the real vector space of all real \(2 \times 3\) matrices, and let \(\mathrm{W}\) be the real vector space of all real \(4 \times 1\) column vectors...
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Let \(a_{i j}\) be the number in the \(i\) th row and \(j\) th column of the array\[\begin{array}{lllll}-1 & 0 & 0 & 0 & \ldots\end{array}\]\(\frac{1}{2} \quad-1 \quad 0 ...
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Suppose(a) \(\mid f(x, y) \leq 1\) if \(0 \leq x \leq 1,0 \leq y \leq 1\),(b) for fixed \(x, f(x, y)\) is a continuous function of \(y\),(c) for fixed \(y, f(x, y)\) is a...
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On the calculator side of the flight computer, time is always found on which scale(s)?A. Outer.B. Inner scale only.C. Inner scale and far inner scale.
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It can be shown that for any three events \(A, B\), and \(C\), the probability that at least one of them will occur is given by\[\begin{aligned}P(A \cup B \cup C)=& P(A)+...
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Find \(\operatorname{gcd}(500,222)\) using euclidean algorithm
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What is the product of 2 and 3?
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What is the dot (Euclidean inner) product of two vectors?
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How do I calculate the dot product of two vectors?
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Calculate $\mathbf{u} \cdot \mathbf{v}$ for the following vectors in $R^{4}$ : $$ \mathbf{u}=(-1,3,5,7), \quad \mathbf{v}=(-3,-4,1,0) $$
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What are the algebraic properties of the dot/inner product?
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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...
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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...
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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\[\langle f, g\rangle...
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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:=...
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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 \(\langle p, q\rangle=\int_{-1}^{1} p(x) q(x) d x\)....
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Let \(\mathcal{P}_{2}\) be the space of quadratic polynomials.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.b) U...
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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 ...
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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 polynomial...
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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 previou...
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Proof or counterexample. Here \(v, w, z\) are vectors in a real inner product space \(H\).a) Let \(v, w, z\) be vectors in a real inner product space. If \(\langle v, w\r...
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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 \(p(x)\)...
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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\ra...
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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)...
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Let \(V, W\) be vectors in \(\mathbb{R}^{n}\).a) Show that the Pythagorean relation \(\|V+W\|^{2}=\|V\|^{2}+\|W\|^{2}\) holds if and only if \(V\) and \(W\) are orthogona...
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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)....
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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.a) Using the...
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Let \(\mathcal{P}_{2}\) be the space of polynomials of degree at most 2 . a) Find a basis for this space.b) Let \(D: \mathcal{P}_{2} \rightarrow \mathcal{P}_{2}\) be the ...
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