0
votes
1answer
23 views

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, ...
1
vote
2answers
48 views

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 ...
0
votes
2answers
25 views

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 ...
1
vote
1answer
28 views

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 $C(X)$ ...
0
votes
1answer
42 views

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 ...
1
vote
2answers
39 views

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 ...
1
vote
1answer
49 views

Inner product question

We are given an inner product of $\mathbb R^3$: $f\left(\begin{pmatrix} x_1\\x_2\\x_3\end{pmatrix},\begin{pmatrix} y_1\\y_2\\y_3\end{pmatrix}\right) = ...
2
votes
1answer
38 views

Prove that set is orthonormal set

In vector space of all real polynomials with inner product $(x,y) = \sum_0^1x(t)y(t)dt$. $x_n(t) = t^n$ for $n = 0, 1, \dots $. Show that functions: $y_0(t) = 1, $ $y_1(t) = \sqrt(3)(2t-1), $ ...
0
votes
0answers
52 views

about vectors norm

in the following article http://blanche.polytechnique.fr/~mallat/papiers/MallatPursuit93.pdf page 3 he say: $$y= \langle y , a_{k_0} \rangle a_{k_0} + R $$ with $a_{k_0}\in D$ with $\forall ...
0
votes
1answer
38 views

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 ...
0
votes
0answers
72 views

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 ...
3
votes
2answers
96 views

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 ...
0
votes
1answer
223 views

Prove projection is self adjoint if and only if kernel and image are orthogonal complements

Let $V$ be an IPS and suppose $\pi : V \to V$ is a projection so that $V = U \oplus W$ (ie $ V = U + W$ and $U \cap W = \left\{0\right\}$) $ \ $ where $U = ker(\pi)$ and $W = im(\pi)$, and if $v = u ...
2
votes
3answers
88 views

Two inner products being equal up to a scalar

I would appreciate a hint on the following problem: Let $V$ be a finite dimensional vector space over $F$. There are two scalar products such that: $$ \forall \ w,v \in V \ \Big(\langle ...
0
votes
1answer
92 views

$l_0$ is all sequences with finitely many non-zero terms. Show $W^\perp=\{y: <x,y>=0, x\in W\}=\{0\}$ where $W = \{x : <x,a>=0\}$.

Consider the inner product space $l_0$ consisting of all infinite sequences of complex numbers with only finitely many non-zero terms, with the inner product of $l^2$ (space of square summable ...
0
votes
4answers
90 views

Why is the inner product not an element of the Hilbert space?

What I know about Hilbert space is that, elements in that space can be complex numbers. But I was confused to read this statement from a book: The inner product, being a complex number, is not an ...
1
vote
4answers
84 views

Inner Product Space on linear transformation on itself

So $V$ is an inner product space and $T : V \to V$ is a linear map such that $$||T(v)|| = ||v||$$ for all $v \in V$. Prove that $$\langle T(v), T(w)\rangle = \langle v, w\rangle$$ for all $v,w \in V$. ...
0
votes
0answers
29 views

$\max_{1 \leq i \leq n}|\langle x,w_i\rangle|$, $\max_{1 \leq i \leq n}|\langle y,w_i\rangle|$ at same $w_i$ if $x$ and $y$ are close enough?

Let $x,y \in \mathbb{C}^n$ with $|x|_2 = |y|_2 = 1$. Let $w_1, \ldots, w_N \in \mathbb{C}^n$. Let $j,k \in \{1,\ldots, N\}$ such that $$ |\langle x,w_j\rangle| = \max_{1 \leq i \leq n}|\langle ...
5
votes
3answers
185 views

Why orthogonal basis?

Lets take the $\mathbb{R}^3$ space as example. Any point in the $\mathbb{R}^3$ space can be represented by 3 linearly independent vectors that need not be orthogonal to each other. What is that ...
0
votes
1answer
143 views

Hoffman & Kunze exercise

This is a problem from Hoffman & Kunze book on linear algebra: Let V be a finite dimensional inner-product vector space. Let U be a self-adjoint unitary linear operator over V. Show that ...
2
votes
3answers
706 views

How do you prove that tr(B^(T) A ) is a inner product?

Consider the vectorspace of all real $m \times n$ vectors and define an inner product $\langle A,B\rangle = \operatorname{tr}(B^T A)$. "tr" stands for "trace" which is the sum of the ...
1
vote
3answers
82 views

A basic question on orthogonal vector

Let $V$ be a finite dimensional vector space and $X$ be a subspace. Let $$\langle u,y\rangle=0 \forall u $$ with the property that $$\langle u,x\rangle =0 \;\forall x \in X$$ where $u,y \in V$. Then ...
0
votes
1answer
61 views

Extrema of a vector norm under two inner-product constraints.

If $\langle\vec{A},\vec{V}\rangle=1\; ,\; \langle\vec{B},\vec{V}\rangle=c$, then: \begin{align} max\left \| \vec{V} \right \|_{1}=?\;\;\;min\left \| \vec{V} \right \|_{1}=? \end{align} Consider the ...
0
votes
2answers
31 views

find scalar product of vectors in rectangular

let us consider following problem and picture we have $ABCD$ rectagular with $AB=3$ and $BC=5$,$F$ and $E$ are midpoints of rectangular sides,we should find scalar product of my question is ...
1
vote
3answers
128 views

Prove an equality between dimensions of kernels

Let $V$ be a inner product space over field $\mathbb{R}$ with $\dim(V)<\infty$, and $T\in \text{Hom}(V,V)$. I'm trying to prove:$$\dim(\ker T)=\dim(\ker T^*)=\dim(\ker TT^*)$$ Also, as a conclusion ...
1
vote
0answers
48 views

A simple piece of a Lemma on Gram-Schmidt

I was looking at a proof of Gram Schmidt theorem and I saw the following lemma, it starts here: First the theorem: if $V$ is an inner product space and $X= \{x_1,\dots, x_n\}$ is a linearly ...
0
votes
2answers
77 views

In an inner product space over $\mathbb R$, prove $ (u,w)=0 \Leftrightarrow \left \| u+w \right \|=\left \| u-w \right \| $

Let $V$ be an inner product space over field $F$ and $u,w\in V$. Prove that if $F=\mathbb{R}$ then: $$ (u,w)=0 \Leftrightarrow \left \| u+w \right \|=\left \| u-w \right \| $$ Is it also true for ...
0
votes
1answer
76 views

polar decomposition on finite dimensional vector spaces

Let $V$ be a finite dimensional inner product space on $\mathbb{F}$ (where $\mathbb{F}$ can be either $\mathbb{R}$ or $\mathbb{C}$) Let $A$ be a linear operator on $V$. The polar value decomposition ...
2
votes
3answers
518 views

Dual space and inner/scalar product space

$V$ is vector space of finite dimension. $〈· , ·〉$ is an inner product on $V$.(Field $F$) We set transformation $T \colon V \rightarrow V^*$ as the following: $(T(v))(w) = 〈v , w〉$. Prove that $T$ ...
3
votes
2answers
53 views

Closed linear subset of a Hilbert space

If $H$ is a Hilbert space, and if $$(a,b)_H=0$$ for every $b \in B \subset H$, where $B$ is a closed linear subset of $H$, does it follow that $a=0$, the zero element of $H$?
2
votes
1answer
144 views

Revisted: What does $R(T^*)^{\perp}$ mean?

A PLACE WHERE I ADD MY THOUGHTS AS I GO: $$w\in R(T^H)^{\perp}:=\{v\in V ~:~ \langle v, T^Hv\rangle =0~\forall~w\in R(T^H)\}$$ $$\langle v , T^Hv \rangle = \langle ? , ? \rangle =0 $$ ...
-3
votes
1answer
87 views

$\operatorname{rank} (T^*) = \operatorname{rank} (T)$ : PROOF

Let $T$ be a linear operator on a finite dimensional inner product space. Prove that $\operatorname{rank}(T^*) = \operatorname{rank}(T).$
4
votes
3answers
128 views

Invertibility in a finite-dimensional inner product space

Let $T$ be an invertible linear operator on a finite-dimensional inner product space. I just want a hint as to how I should prove that $T^{\star}$ is also invertible and $( T^{-1} )^{\star} = ( ...
1
vote
4answers
191 views

Proof: $\det\pmatrix{\langle v_i , v_j \rangle}\neq0$ $\iff \{v_1,\dots,v_n\}~\text{l.i.}$

Let $V$ be a real inner product space and $S=\{v_1,v_2, \dots, v_n\}\subset V$. How am I to prove that $S$ is linearly independent if and only if the determinant of the matrix $$ ...
1
vote
1answer
85 views

Hermitian Inner Product | Basis | Orthogonal Complement

If I say $X = \{x, x'\}\subset\mathbb{F}^3$ is a subspace, where $x$ and $x'$ are linearly independent (for some field $\mathbb{R}$ or $\mathbb{C}$), with $$\mathbb{F}^n := ...
-9
votes
2answers
208 views

$\mathbb{C}^3$: 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 ...
3
votes
3answers
106 views

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 ...
0
votes
1answer
54 views

Inner product Proof,

So $V$ is an inner product space (finite dimensional) with inner product defined. If $v$ and $w$ are vectors in $V$, how would one go about proving this? $\langle \phi_\beta (x), \phi_\beta (y) ...
0
votes
2answers
69 views

I need a proof for a scalar product - no numbers allowed

Hello I simply cant explain to myself why this equation holds. Lets say we have an orthonormal basis $\vec{i}, \vec{j}$ and 2 2-D vectors in this basis which are: \begin{align} \vec{a} &= ...
0
votes
2answers
458 views

Calculate distance from plane to parallel plane in O using vector and normal

I'm trying to figure out what's the best method to get the distance between two planes where i have the normalized vector of the plane and a point in the plane. What I want to do is to create a ...
2
votes
1answer
95 views

On affine spaces, distances, angles, and coordinates

Let's define an affine space as a pair $(A, V)$, where $A$ is a set and $V$ is a vector space, together with a map $V\times A \rightarrow A, \;\; (v, a) \mapsto v + a,$ such that $\forall \, a \in ...
0
votes
1answer
134 views

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 ...
3
votes
4answers
163 views

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 ...
4
votes
1answer
75 views

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 ...
0
votes
1answer
50 views

Kernels of Adjoints

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 ...
0
votes
1answer
371 views

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\\ ...
4
votes
2answers
59 views

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 ...
2
votes
2answers
165 views

True or False: Every finite dimensional vector space can made into an inner product space with the same dimension.

Every finite dimensional vector space can made into an inner product space with the same dimension.
1
vote
1answer
204 views

Inner product for a finite dimensional vector space

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.
0
votes
2answers
174 views

what is $ M^{\perp}$ given set?

Let $ ‎X=C[-1,1]‎$‎‎ be inner product space with definition $$‎\langle f,g‎‎‎\rangle =‎\int_{-1}^1 f‎‎ \overline{g}‎ ‎dt ‎‎.$$ Let $M$ be the subspace defined by ‎$$ ‎M= ‎‎\left\{f‎ \in ‎X\mid ...