# 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?
510
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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 $$y$$.
414
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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... 194 views 0 answers 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 $$m, n \in \mathbb{N}$$ 333 views 1 answers How do I round real numbers to an appropriate degree of accuracy? 585 views 1 answers 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... 292 views 1 answers 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 ... 222 views 1 answers 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?
557
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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... 499 views 0 answers 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:=... 375 views 0 answers 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$$ are the quadratic polynomial...
486
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In a complex vector space (with a hermitian inner product), if a matrix $$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... 527 views 0 answers 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... 497 views 0 answers 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)... 462 views 0 answers 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$$ are orthogona...
Let $$W$$ be a linear space with an inner product and $$A: W \rightarrow W$$ be a linear map whose image is one dimensional (so in the case of matrices, it has rank one)....
Let $$\mathcal{P}_{3}$$ be the space of polynomials of degree at most 3 anD let $$D: \mathcal{P}_{3} \rightarrow \mathcal{P}_{3}$$ be the derivative operator.a) Using the...
Let $$\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 ...