# Tagged Questions

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### V = U⊕W then Prove that (V/W)* is isomorphic to W^0

Let $V$ be a vector space (not necessary being finite dimensional) and let $U$, $W$ be subspaces of $V$ such that $V = U\oplus W$. Prove that $(V/W)^*$ is isomorphic to $W^0$. note: (V/W)* is the ...
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### Inner product over the $C^2$

Let a, b, c, d ∈ C and consider the vector space $C^2$ Suppose inner product is defined as: $⟨x, y⟩ = ax_1\bar y_1 + bx_2\bar y_1 + cx_1\bar y_2 + dx_2\bar y_2$ I am trying to find all a, b, ...
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### Hilbert vs Inner Product Space

What is the difference between a Hilbert space and an Inner Product space? They both seem to be defined as simply a vector space equipped with an inner product. Also can a metric always be defined ...
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### inner product space definition

I have some problem in the definition of inner product space. The book I use to learn in linear algebra and its application 4th edition (David C.Lay) In the chapter 6.7 it define the inner product ...
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### Are orthogonal spaces exhaustive, i.e. is every vector in either the column space or its orthogonal complement?

Quick question about subspaces, just to make sure I have this straight in my head. Taking an $n\times k$ matrix X with $rank(X)=k$, is every vector in $\mathbb{R}^n$ in either the column space $C(X)$ ...
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### Equivalent definitions of isometry

Consider a map $T:\mathbb{R}^2\to\mathbb{R}^2$ such that $\lVert T(x)\rVert=\lVert x\rVert$. Is this equivalent to stating that $\langle x, y\rangle=\langle T(x), T(y)\rangle$ for all ...
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### For inner product spaces, do we have $||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||$?

Let $V$ be an inner product space. Then for all $\vec{u},\vec{v} \in V$ we have $$||\vec{u}-\vec{v}|| \leq ||\vec{u}||+||\vec{v}||.$$ I know that the converse to the equation is true such that ...
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### Proof that two spans are equal

I have an orthogonal subset of nonzero vectors $u_1,\dots,u_n$. I take $v \in V$ (a vector space) so that $v$ is the in the span of the previous set. Now I let $u$ be $v-m_1u_1-\dots-m_nu_n$ with ...
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### What does it mean for a vector space to 'have' a particular inner product?

If a vector space 'has' a particular inner product, what does that mean exactly? Is it that all the vectors in the space satisfy the conditions for the inner product to work? Is it that all the ...
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### Dual spaces and inner product

What is the relation (if any) between dual spaces and inner product? As far as I understand the dual space of a vector space is the set of all linear mappings from the vector set to the field over ...
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### $\mathbb{C}^3$: Orthogonal Complement

Let $S=\{(1,0,i),(1,2,1)\}$ in $\mathbb{C}^3$. What is the method used to find a basis for $S^{\perp}$? EDIT$^1$: I think this bit of literature from Gockenbach's Finite-Dimensional Linear Algebra ...
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### Multipliciousness within an inner product space.

Question: Let $V$ be an inner product space and $v,w\in V$. Prove that $\lvert\langle v,w\rangle\rvert=\lVert v\rVert \lVert w\rVert$ if and only if one of the vectors $v$ or $w$ is a multiple of ...
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### 3 equations with 9 unknown variables with scalar product

Excuse my bad english pls. I can't find a proper solution to my problem because i don't know the exact mathematical terms in english. My problem is how to get the 3 elements of each of 3 vectors ...
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### If $X$ is an orthogonal matrix, why does $X^TX = I$?

It's not immediately clear to me why this is true. My notes say that putting $n$ orthonormal vectors $v_1, ..., v_n$ in the columns of $X$ gives $X^TX = I$, and it follows from this that the rows of ...
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### Inner product spaces that are isometrically isomorphic

I know this is a fundamental result in linear algebra, and although it is referenced in my textbook, it does not have a proof for it. I was wondering if someone could help me out: Let $V$ and $W$ be ...
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Let $A$ be an $m \times n$ matrix. Show that $\mbox{Ker} A = \mbox{Ker} (A^*A)$. To do that you need to prove 2 inclusions, $\mbox{Ker} (A^*A)$ is a subset of $\mbox{Ker} A$ and $\mbox{Ker} A$ is a ...
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### How to verify this is an orthogonal basis? How to transform it into an orthonormal basis?

Let B = \left\{ \begin{bmatrix} 3\\ -3\\ 0\end{bmatrix},\begin{bmatrix} 2\\ 2\\ -1\end{bmatrix},\begin{bmatrix} 1\\ 1\\ 4\end{bmatrix}\right\},\qquad v =\begin{bmatrix} 5\\ -3\\ ...
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### Another linear algebra question

I have no idea how to start the following question. Any help will be greatly appreciated. (a) Let $A$ be a $n\times n$ matrix and let $a_1,...,a_n$ be the rows of $A.$ Suppose $y=(y_1, ..., y_n)$ is ...
Do we always have an inner product for a finite dimensional vector space $V$ over a field $k$ such that $V$ is a Hilbert space? Thank you very much.