# Tagged Questions

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### What does it mean that the product of two vectors produces real number?

I am going over inner product space. I know that linear space has an inner product as long as it satisfies $4$ conditions. And, the book says that for $x,y$ in $V$, there is a real number ...
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### Need verification - Given a Hermitian matrix and two eigenvectors corresponding to distinct eigenvalues, show x and y are orthogonal.

Claim: Let $A \in \mathbb{C}^{mxm}$ be hermitian ($A = A^*)$. If $x$ and $y$ are eigenvectors corresponding to distinct eigenvalues, then x and y are orthogonal. Proof: Let $x$ and $y$ correspond to ...
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### What does "$S\times S\rightarrow R$

In the textbook, Mathematical Methods and Algorighms for Signal Processing, Tood K. Moon, the $\mathbf{inner\;product}$ is defined it is a function $\langle\cdot,\cdot\rangle:S\times S\rightarrow R$ ...
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### Showing the conjugate symmetric property of an inner product when we don't know if our field is $\mathbb{C}$.

The conjugate symmetric property of an inner product states that $\langle{x, y}\rangle = \overline{\langle{y, x}\rangle}$. My question is regarding showing this when we don't necessarily know that our ...
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### Fourier coefficients with respect to an orthonormal basis for an inner product space

$V = \operatorname{span}(S)$, where $S = \{(1, i, 0), (1 - i, 2, 4i)\}$, and $x = (3 + i, 4i, -4)$. Apply the Gram–Schmidt process to the given subset $S$ of the inner product space $V$ ...
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### If $\|Tv\|=\|T^*v\|$ for all $v\in V$, then $T$ is a normal operator

I have solved a question but I am not sure the last step of the question. If someone can verify it that would be great. Let $V$ be a finite dimensional vector space with complex inner product. Let ...
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### Calculate the angles between $(1,X),(X,X^2),(X^2,X^3),(X^3,X^4)$ given the inner product $\langle p(x),q(X) \rangle = \int_{-1}^{1} p(X)q(X)dX$

Let $V_4$ be the vector space of all polynomials of degree less than or equal to 4 with the inner product $$\langle p(x),q(X) \rangle = \int_{-1}^{1} p(X)q(X)dX$$ calculate the angles between ...
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### Let $\langle x,y \rangle = x_1y_1 + 3x_2y_2 + 4x_3y_3 + x_1y_2 + x_2y_1 + x_1y_3 + x_3y_1 + x_2 y_3+x_3y_2$, prove pos. definiteness

Let $$\langle x,y \rangle = x_1y_1 + 3x_2y_2 + 4x_3y_3 + x_1y_2 + x_2y_1 + x_1y_3 + x_3y_1 + x_2 y_3+x_3y_2$$, prove that $\langle x,y \rangle$ is positive definite. I have simplified this to the ...
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### Using inner product property to determine if operator is an isomorphism.

Let $\varphi$ be an operator on a $k$-vector space $V$ with an inner product $\langle\cdot,\cdot\rangle$. Suppose that $\langle v,\varphi v\rangle = 0$ for every $v\in V$. If we take $k=\mathbb R$, is ...
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### Let $G_1 = \{v_1 + \lambda w_1 | \lambda \in \mathbb{R}\}, G_2 = \{\{v_2 + \mu w_2 \}$ be two skew lines, derive a formula for $d(G_1,G_2)$

Let $G_1 = \{v_1 + \lambda w_1 | \lambda \in \mathbb{R}\} \subseteq \mathbb{R}^n, G_2 = \{\{v_2 + \mu w_2 |\mu \in \mathbb{R}\}\subseteq \mathbb{R}^n$, with $v_1, v_2, w_1, w_2 \in \mathbb{R}$ be two ...
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### self adjoint transformations and inner product problem

Let $(V, <,>)$ be a finite dimensional vector space with an inner product, and let $f,g \in End(V)$ two self adjoint linear transformations. (a) Prove that if $f$ and $g$ commute, then, for ...
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### Angle between two polynomials

Given the inner product of two polynomials $p(X), q(X) \in P(d)$, where $P(d)$ is the vector space of all polynomials of degree less than or equal to d, with real coefficients, and using the inner ...