0
votes
1answer
22 views

Which of the following expresses the fact that the vectors $u$ and $v$ have the same length?

Which of the following expresses the fact that the vectors $u$ and $v$ have the same length? (a) $u · u = v · v$ (b) $||u + v|| = ||u|| − ||v||$ (c) ${u \over||u||} = {v \over ||v||}$ (d) $||u + ...
2
votes
2answers
37 views

Is $u^TAu \geq 0$ true for all symmetric matrices $A$?

we know from the definition of inner product and norm, that $u^Tu$ is always larger than zero, except the case where $u=0$ at which case it is zero. I came across a question that infers that $u^TAu ...
0
votes
1answer
12 views

Showing equivalence of the positivity condition of inner products

I have no idea how to prove this, first off, because I don't think I understand the question. Isn't the second case not true for v = 0. Show that for real vectors spaces $V$ with $V$ $\not= {0}$, ...
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, ...
2
votes
2answers
41 views

Why $\langle y,x\rangle+\langle x,y\rangle=2\mathrm{Re}\langle x,y\rangle$? And the rules of using absolute value, inner production and norm?

Let V be an inner product space over F, x,y∈V. In the proof of triangle inequality, my textbook uses $$\|x+y\|^2 = \langle x,x \rangle + \langle y,x \rangle + \langle x,y \rangle + \langle y,y \rangle ...
1
vote
1answer
24 views

Prove triangle inequality of vector norm

I am trying to show that $||x+y||_p \leq ||x||_p + ||y||_p$ where $p$ is an integer larger than 1, but not infinity (I proved those cases already), and $||x||_p = (\sum_{i=1}^n |x_i|^p)^{\frac{1}{p}}$ ...
1
vote
1answer
33 views

Some kind of projection in a non-orthogonal basis

Sorry if the title sounds convoluted, I couldn't find any better. In $R^d$, let $(e_1,\ldots, e_d)$ be a basis. Show there exists $(a_1,\ldots, a_d)$ d vectors of $R^d$ such that $$\forall x \in ...
0
votes
1answer
24 views

$T$ is a linear operator on a IPS $V$ which has a basis $\beta$. Prove that $A_{ij} = \langle T(v_j),v_i \rangle$

I have trouble understanding a proof on textbook and I would appreciate your help! Corollary. Let $V$ be a finite-dimensional inner product space with an orthonormal basis $\beta = \{v_1, v_2, ...
1
vote
0answers
33 views

Linear Algebra quick Question over inner product space

In an inner product space, not necessarily $\mathbb R^n$, there are vectors $a$ and $b$ such that $||a||\cdot ||b|| < |\langle a,b\rangle| $ Is this never true?
0
votes
2answers
34 views

Linear Algebra Explanations on true and false.

1.Could someone prove that if a set of vectors in a $p$-dimensional vector space $Q$ is a spanning set for $Q$, it is a basis. 2.If $T$ is a linear transformation from $\mathbb R^3$ onto $P_2$, then ...
0
votes
0answers
30 views

Quick Question on showing a function is an inner product

I just have a quick question How come =p(1)q(1)+p(2)q(2) is an inner product but =p(1)q(1)+p(2)q(2)-p(3)q(3) is not?
0
votes
1answer
29 views

Linear Algebra Quick Question on orthonormal basis and inner product

I have a question asking to find an orthonormal basis of $p_2$ with respect to the inner product =2 X integral from 0 to 1 p(x)q(x)dx. What do I do with the 2 in front of the integral? When I solve ...
0
votes
1answer
35 views

Finding an orthonormal basis for the space $P_2$ with respect to a given inner product

I am so confused on what to do for this question. The questions asks to find an orthonormal basis of $P_2$, the space of quadratic polynomials, with respect to the inner product $$ \langle p, ...
3
votes
1answer
55 views

If $\|u\| \leq \|u+av\|$ for all $a \in F$, How can I show that $\langle u,v\rangle=0$?

If $\|u\| \leq \|u+av\|$ for all $a \in F$, How can I show that $\langle u,v\rangle=0$? I know a standard solution uses $\operatorname{Re}$ and $t =$ something but was wondering if there was ...
0
votes
1answer
41 views

Finding orthogonal complement

Let $X$ be an inner product space and let $x\in X$. $M=\{z\in X:\langle z,x\rangle=0\}$. I want to find $M^{\perp}$ and $M^{\perp \perp}$. Clearly, $\{x\}\subset M^\perp$. Thus $M^{\perp ...
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 $C(X)$ ...
2
votes
1answer
56 views

Inner-product question

Let $V$ be $\mathbf{R}^2$ equipped with usual inner product, and $v$ be a nonzero vector. $S_v(u)= u- 2 \frac{\langle u,v\rangle}{\langle v,v\rangle } v$ and $\Phi$ be a non-empty set of unit vectors ...
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
1answer
23 views

Prove that orthogonality in Euclidean space is geometrically perpendicularity?

Is this simply true by definition (that is, taken as axioms?) How would one to prove that for $||\vec{x}||=1$ and $||\vec{y}||=1$, if $(\vec{x},\vec{y})=0$, then $\vec{x}\perp\vec{y}$? In other ...
2
votes
1answer
100 views

$||u||\leq ||u+av|| \Longrightarrow \langle u,v\rangle=0$

Prove that $\langle u,v\rangle=0\Longleftrightarrow ||u||\leq ||u+av||$. So far I can get the $\Longrightarrow$ very easily, but I need some help with the $\Longleftarrow$ implication, any hints ...
3
votes
3answers
72 views

Necessity of completeness of the inner product space in Riesz representation theorem

I wanted to find a counter example to show that the completeness of the inner product space is necessary in Riesz representation theorem. Please give an example of a bounded linear functional $T$ on ...
4
votes
1answer
53 views

Show that $T\neq{T^*}$

Let $V=P_2(\mathbb{R}), T\in \mathcal{L}(P_2(\mathbb{R})),$ where $T(p)=(a_1x)$. Make $V$ an inner product space by defining $$\langle p,q\rangle=\int_0^1{p(x)q(x)\,dx}$$ So I calculate $$\langle ...
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 ...
2
votes
1answer
41 views

Show a linear transform is self adjoint - check my answer

We are given $T:V \to V$ a normal linear transform (meaning $TT^*=T^*T$) We are also given $T^2=T$. Show that $T$ is self adjoint (meaning $T^*=T$). What I did I think I may have done something ...
2
votes
2answers
105 views

$T^*T=TT^*$ and $T^2=T$. Prove $T$ is self adjoint: $T=T^*$ [duplicate]

$V$ is an inner product space of finite dimension over $\mathbb{R}$, and $T:V\to V$ a linear transformation which is normal, that is, $T^*T=TT^*$. In addition $T^2=T$. Prove $T$ is self adjoint, 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) = ...
1
vote
1answer
17 views

Matrix Composed of Traces of Linear Independent Set of Matrices

I got the following problem: Let $S=\{A_1,A_2,...,A_k\} \subseteq \mathbb{M^R}_{n\times n}$ be a linear independent set of $k$ real $n \times n$ matrices with respect to the standard matrix inner ...
2
votes
2answers
42 views

$T: \mathbb R^n \to \mathbb R^n$, $\langle Tu,v\rangle=\langle u,T^*v\rangle$, is $T^*=T^t$ regardless of inner product?

Basic question in linear algebra here. $T$ is a linear transform from $\mathbb R^n$ to $\mathbb R^n$ defined by $T(v)=Av$, $A\in \mathrm{Mat}_n(\mathbb R)$. We are given some inner product $\langle ...
2
votes
1answer
36 views

Orthogonality on complex inner product space

Let $V$ be a complex inner product space. I need to show the following: $(x\ and \ y\ are\ orthogonal)\ \Rightarrow (\left \| \lambda x+\beta y \right \|^{2}=\left | \lambda \right |^{2}\left \| x ...
4
votes
2answers
41 views

What am I doing wrong? Gram Schmidt process..

Let there be the inner product of all polynomials of degree smaller or equal to 2: $\langle f,g\rangle=\int_0^1f(x)g(x)xdx$. Find orthonormal basis. So I really tried this for an hour and it pretty ...
1
vote
1answer
44 views

Does this function define an inner product?

Does the function below define an inner product? $$\langle (x, y), (z, t)\rangle = xz − yt$$ I know how to prove it given two vectors (e.g. $\langle(x,y),(z,t)\rangle$) demonstrating symmetry, ...
3
votes
3answers
75 views

Gram-Schmidt for uncountable sets?

I know that Gram Schmidt can be applied to countable linear independent sets on Hilbert spaces, but what happens if we apply it on uncountable sets? Obviously this process has to fail then (at least ...
0
votes
1answer
27 views

normalize a vector in $\mathbb C^3$ - a very basic question

I think I forgot a bit previous-year Linear Algebra, so I have a very basic question to you. Given the following question: Normalize the following vector: $v \in \mathbb {C^3}, \space v = i, -i, ...
0
votes
0answers
51 views

Linear form, there exists a unique vector $z$ such that $f(w)=\langle w,z \rangle$

Let $V$ be a finite dimensional space over the field $\mathbb{F}$ with inner product $\langle \cdot, \cdot \rangle$. Then for every linear form $f: V \rightarrow \mathbb{F}$ there exists a unique $z ...
0
votes
2answers
58 views

Orthogonal complement of a single vector

Is there a quick way to show that if v is an element of an n dimensional real inner product space space, the orthogonal complement of v is n-1 dimensional? I can do this by using gram-schmidt and ...
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
1answer
20 views

Question about definite positive symmetric bilinear form

Let $A$ be the matrix of a positive definite symmetric bilinear form. Prove $a_{11}a_{nn}\ge a_{1n}a_{n1}$. I don't really have a clue of how to solve this.
1
vote
0answers
66 views

Every inner product space is a normed space which is also a metric space

As I am pretty sure that everybody knows that a Hilbert space is a space that is a complete, separable and generally infinite dimensional inner product space. By the means of completion, every Cauchy ...
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 ...
0
votes
2answers
83 views

General Proof that $\left< v, v \right> = \left|\left| v \right|\right|^2$

Consider $\mathbb{R}^n$ with the standard Euclidean inner-product. I'm trying to give a proof that $$ \left< v, v \right> = \left|\left| v \right|\right|^2 $$ but can only seem to do it for ...
1
vote
1answer
63 views

Must unitary matrices satisfying this property commute?

If A and B are unitary matrices such that A, B, and AB are all conjugate to diag(1,1,-1,-1), must AB=BA? Why or why not?
0
votes
0answers
39 views

Adjoints and Inner Product Spaces

The proposition states that if $V$ is a finite dimensional IPS and $\ \phi : V \rightarrow V$ is a linear operator, then $\phi$ has an adjoint $\phi^*$ and if the matrix of $\phi$ wrt to an ...
-1
votes
1answer
44 views

Find conjugate transpose of linear transform

A difficult question I've been trying to tackle but I seem to hit a dead end. let $V$ be an inner product space over $\mathbb R$. We are required to find $T^{*}$ such that $<T(u),v> = ...
1
vote
3answers
51 views

Prove a space is Hilbert [duplicate]

I got stucked in this problem and get no clue to solve this. Can any one please help me? Thanks Suppose $X$ is an inner product space. If for every bounded linear function $f$, there exists $z \in ...
2
votes
1answer
161 views

prove that if $\lambda$ is an eigenvalue of T then $\bar\lambda$ is eigenvalue of $T^*$

I have to prove that if $\lambda$ is an eigenvalue of T then $\bar\lambda$ is eigenvalue of $T^*$ (adjoint) I know that $<Tv,u> = <\lambda v,u> = ...
2
votes
0answers
25 views

Linear group action over an hermitian space.

Let $\mathrm{h}$ be an non degenerate hermitian form on $\mathbb{C}^2$, and suppose $\mathrm{h}$ is of signature $(1,1)$. Let $A$ and $ B \in \mathrm{SL}(2,\mathbb{C})$ such that $G$ the group ...
5
votes
1answer
67 views

Prove, that f is a linear map.

$U,V$ - Euclidean spaces $f:U \rightarrow V$ $f(0)=0$ $ \forall _{u,v \in U}:d(f(u),f(v))=d(u,v)$ Prove that $f$ is a linear map. I'm thinking about something like this: $||f(u+v)|| =d(f(u+v),0) = ...
1
vote
2answers
29 views

Prove that [B]ij = <ui, uj>

Let B = A*A. That is, the product of a matrix and its transpose. Prove that the $[B]_{ij} = < u_{i}, u_{j}>$. Now, upon looking at this, I decide to test it out first to see if I believe it. I ...
0
votes
1answer
41 views

linear algebra: inner product

6. Consider $V = \def\P{\mathbb P^2}\P$ with inner product: $$\def\sp#1{\left<#1\right>}\sp{p(X), q(X)} = 2p(-1)q(-1) + 3p(1)q(1) + p(2)q(2) $$ a. Show that for any non-zero polynomial ...
2
votes
1answer
29 views

Inner product problem

I'm having trouble with this excercise: Determine the orthogonal space of odd continuous functions in the space of continuous functions. (In the interval $ [-1,1] $) I assume that the inner product ...