0
votes
0answers
20 views

Minimum in complex inner product vector space

I'm stuck at this problem, can someone give me a hint? Let $x_i$ and $y_i$ ($i=\overline{1,n}$) be vectors in an infinite dimensional vector space $V$ with inner product $(,)$ satisfy: ...
0
votes
1answer
35 views

Few basic things unclear to me about inner product spaces and orthonormal basis

Few things unclear to me about inner product spaces: assume V is an inner product space with B orthonormal basis. Why is it true that: $$\langle x,y\rangle = \langle[x]_{B} , [y]_B \rangle{st}$$ ...
0
votes
1answer
33 views

Find the signature of $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$

In $\mathbb{R}^n$ let $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$ quadratic form. $a:=(a_1,\ldots,a_n)\neq0$ $\in \mathbb{R}^n$ find the signature of $Q$
1
vote
2answers
49 views

Angle between two vectors on manifold

I'm parallel transporting a vector along a curve and trying to calculate how much this vector rotates relative to the curve's tangent vector. So if the path is a geodesic then I will get an answer of ...
1
vote
1answer
85 views

How to prove $ |\langle u,v\rangle| \leq ||u||||v||$

How to prove $ |\langle u,v\rangle | \leq ||u||||v||$ Note: I have given this many attempts so don't downvote due to lack of effort, refer to edit history for evidence of said effort
0
votes
3answers
135 views

Subspace of a finite dimensional inner product space, independence of basis choice

Let $W$ denote a subspace of a finite dimensional inner product space $V$, and let $$\beta = \{w_1,w_2,\dots,w_r\}$$ denote an orthogonal basis for $W$. For any $v\in V$ define $$proj_{\beta}v = ...
0
votes
2answers
23 views

Scalars determine a vector in inner product space.

Let $V$ be a finite dimensional inner product space over $k$ with basis $\{v_1,\dots,v_n\}$ and inner product $\langle \cdot,\cdot\rangle$. For any $\alpha_1,\dots,\alpha_n\in k$, there exists a ...
0
votes
2answers
42 views

Why are these vectors expressed as row vectors and not column vectors? When to write as row vectors or column vectors?

Everytime I have been asked to find a basis when the vectors were given in comma delimited form, I, and the book, would write out the vectors as columns in a matrix. Another example in the book ...
0
votes
1answer
25 views

Prove: given a basis of vector space, we can find an inner product such that this basis is orthonormal

$V$ is vector space above fields $\mathbb{R}$ or $\mathbb{C}$ , and $B = \{v_1,...,v_n\}$ is a basis of $V$. I need to prove that there is an inner product on vector space $V$, such that $B$ is an ...
0
votes
0answers
28 views

Custom Norm Function Proof $\left \| x \right \|=\left | \xi _{1} \right |+\left | \xi _{2} \right | $

For Vector Space X consisting of ordered pairs of Complex numbers, Can we define the Norm stated below from inner product, in X ? $$\left \| x \right \|=\left | \xi _{1} \right |+\left | \xi _{2} ...
5
votes
1answer
76 views

Is it possible to define an inner product such that an arbitrary operator is self adjoint?

Given a vector space $V$ (possibly infinite dimensional) with inner product $(.,.)$. We say an operator $A$ is self adjoint if $(Af,g)=(f,Ag)$. The definition as stated require us to start with an ...
0
votes
1answer
28 views

Linear Algebra,Conjugate Transpose

Let $ M_n(\mathbb C) $ be the space of all $ n\times n $ matrices with complex entries. Prove that function $ \langle, \rangle : M_n(\mathbb C) \times M_n(\mathbb C) \to \mathbb C $ defined by $ ...
2
votes
1answer
99 views

Different Types of Inner Products in R^n

I know that in $\mathbb{R}^n$ (and in general $F^n$) you can define an Inner Product in the following way: Let $X,Y \in \mathbb{R}^n$. Let $C \in \mathbb{R}^n$, and let all the components of $C$ be ...
1
vote
1answer
58 views

Prove that the mapping $\psi : L(V,W) \rightarrow L(W^*, V^*)$ given by $\psi(T) = T^t$ is an isomorphism.

Let $V,W$ be finite-dimensional vector spaces over the same field $\mathbb{F}$ and let $L(V,W)$ be the vector space of $\mathbb{F}$-linear transformations from $V$ to $W$. Prove that the mapping ...
0
votes
1answer
84 views

inner product space , dual space, proof about isomorphism

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 $V^\ast/(W^0)$ is isomorphic to $W^\ast$. Notation and ...
0
votes
1answer
43 views

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 ...
0
votes
1answer
27 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
65 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
51 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
44 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
45 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
40 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
56 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
48 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
49 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
75 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
155 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
426 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
100 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
97 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
100 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
101 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
30 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
319 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
153 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
942 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
85 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
68 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
34 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
134 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
49 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
81 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
88 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
644 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
58 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
154 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
92 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
157 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
210 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 $$ ...