0
votes
0answers
38 views

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 ...
1
vote
1answer
21 views

Finding values for which a bilinear form is an inner product

I am trying to find the values (if any) of p and q for which the following satisfies the definition of an inner product: $$ \left \langle \mathbf{z}, \mathbf{w} \right \rangle = z_1\overline{w_1} + ...
0
votes
1answer
62 views

Concepts of isomorphisms of linear spaces with a norm and inner product

If I have a topological space, I say that a homeomorphic map preserves the structure of this space. Thus, in order to preserve topological properties we want to have a continuous bijection with a ...
2
votes
2answers
58 views

Find the $L^2[-\pi,\pi]$ projection of $f(x)$

I need to find the $L^2[-\pi,\pi]$ projection of $f(x)=x^2$ onto the space $V_n\subset L^2[-\pi,\pi]$ spanned by ...
1
vote
0answers
37 views

Variation of orthogonal vectors

It is given that inner product $$ \left\langle a(t),b(t)\right\rangle =0,\quad \forall t\in[0,T] $$ where $a(t), b(t)\in \mathbb{R}^n$. If $\dot{a}(t)$ is known, is there a way to find an expression ...
2
votes
2answers
35 views

Clarification on Some Definition of Inner Product Space

Suppose $V$ is finite-dimensional Real vector space and $T\in \mathcal{L}(V)$. Suppose that $V$ has a basis $(e_1,e_2,\ldots, e_n)$ of eigenvectors of $T$, every element of $V$ can be written as a ...
1
vote
2answers
29 views

Prove that function is inner product

$V$ is a space of polynomials, we have $p=a_0+a_1x+\dots +a_nx^n$ og $q=b_0+b_1x+\dots +b_nx^n$. I need to show this function is an inner product: $$\langle p,q\rangle=\sum_{j=0}^n ...
0
votes
2answers
29 views

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 ...
0
votes
2answers
22 views

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 $\langle\cdot,\cdot\rangle:S\times S\rightarrow R$ ...
0
votes
0answers
26 views

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 ...
1
vote
1answer
30 views

Let $A$ be an invertible , prove that there exist positive constants $c_1$ and $c_2$ such that $c_1X^tX\leq X^tA^tAX\leq c_2X^tX $

Let $A$ be an invertible $n \times n$ matrix over $\mathbb{R}$, prove that there exist positive constants $c_1$ and $c_2$ such that $$c_1X^tX\leq X^tA^tAX\leq c_2X^tX $$ for all $X \in ...
4
votes
1answer
53 views

Does $S^\bot+T^\bot = (S\cap T)^\bot$ hold in infinite-dimensional spaces?

If $S$ and $T$ are subspaces of some finite-dimensional inner product space then $$S^\bot+T^\bot = (S\cap T)^\bot.$$ See, for example, this post or this post Does it hold in infinite-dimensional ...
4
votes
2answers
58 views

Let $W_1$ and $W_2$ be subspaces of a finite dimensional inner product space space. Prove that $(W_1 \cap W_2)^\perp=W_1^\perp + W_2^\perp $

Let $W_1$ and $W_2$ be subspaces of a finite dimensional inner product space space. Prove that $$(W_1 \cap W_2)^\perp=W_1^\perp + W_2^\perp $$ My Try One direction is easy : Let $\alpha \neq 0$ ...
2
votes
0answers
40 views

Show that $(Au,Bv)=(u,A^tBv)$

Let $ A, B $ be matrices of order $ n $, and $ \vec{u}, \vec{v} $ vectors from euclidean space $ \mathbb{R}^n $, then $ (Au,Bv) = (u,A^tBv) $ pd. $(\cdot ,\cdot)$ is my notation for inner product, ...
3
votes
1answer
86 views

Question about existence of unitary matrices with certain properties

We are given a set of $d$ normalized vectors on a $d$-dimensional complex vector space: $e_1$, $e_2$... $e_d$, where $$\langle e_j,e_j\rangle=1$$ for all $j$. These are not necessarily mutually ...
0
votes
1answer
36 views

How can I prove that the span of an a subspace and it's orthogonal complement is the whole vector space?

The book Linear and Geometric Algebra explains the following theorem in a way that I haven't been able to figure out: If $\mathbf{A}$ and $\mathbf{B}$ are subspaces of a vector space $\mathbf{B}$ ...
0
votes
1answer
43 views

Find the orthogonal projection of $f(x)=4x^2−4$ onto the subspace spanned by $g(x)=x−12$ and $h(x)=1$.

Use the inner product $\langle f,g\rangle =\int_0^1 f(x)g(x)dx$ in the vector space $C^0[0,1]$ to find the orthogonal projection of $f(x)=4x^2−4$ onto the subspace $V$ spanned by $g(x)=x−1/2$ and ...
1
vote
2answers
39 views

How to find orthonormal basis for inner product space?

In $\mathbb{R}^3$ we declare an inner product as follows: $\langle v,u \rangle \:=\:v^t\begin{pmatrix}1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 3\end{pmatrix}u$ How can I find an ...
1
vote
2answers
52 views

question about inner product and $f^*$

In $\mathbb{R}$3 we declare an inner product as follows: $\langle v,u \rangle \:=\:v^t\begin{pmatrix}1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 3\end{pmatrix}u$ we have operator $f ...
1
vote
2answers
46 views

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 $V$ ...
2
votes
2answers
65 views

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 ...
0
votes
1answer
92 views

Project sin(x) onto orthonormal basis that span ${(1, x, x^2, x^3, x^4, x^5)}$ on domain $[-\pi, \pi]$

I am self-studying LA through Linear Algebra Done Right 2nd ed. I probably made a blatant error somewhere but I have been stuck for a whole day now. The book gave the answer $0.987862x − 0.155271x^3 ...
0
votes
0answers
44 views

Defining an inner abstract vector space

Since an inner product space is an abstract vector space with an additional structure called an inner product, and this additional structure is a component wise operation that associates each pair of ...
1
vote
2answers
60 views

Is a norm on $R^n$ linear?

I was reading the book Linear Algebra Done Right by Axler. In the chapter on inner product space (Ch.6), he defines the norm of x on $R^n$ space as: $||x|| = \sqrt{x_1^2 + ... + x_n^2}$ and says: ...
0
votes
1answer
58 views

Why $\langle a,x\rangle = \langle b,x\rangle,\forall x\in X\implies a=b$ [closed]

Let $X$ be (possibly infinite-dimensional) Hilbert space. How can we show that if $$\langle a,x\rangle = \langle b,x\rangle,\forall x\in X$$ then $a=b$?
0
votes
3answers
49 views

Rigid motion on $\mathbb{R}^2$ which fixes the origin is linear

Let $V=\mathbb{R}^2$ be an inner product space with the standard inner product, and let $T$ be a rigid motion of $V$. Suppose $T(0)=0$, prove that $T$ is linear. (A rigid motion of an inner product ...
5
votes
3answers
55 views

Area preserving transformation in a higher dimensional space is unitary.

In $\mathbb{R}^3$, a linear operator $Q:\mathbb{R}^3 \to \mathbb{R}^3$ preserves the area of parallelograms: that is, given $x,y\in \mathbb{R}^3$, the area of a parallelogram formed by $x$ and $y$ is ...
1
vote
2answers
63 views

Show that $\langle x, Ax \rangle + \langle b, x \rangle = c$ can be transformed to $\langle x', Ax' \rangle = 1$

Let $A$ be a real, regular, symmetric $n \times n$ matrix, $b \in \mathbb{R}^n$ and $c \in \mathbb{R}$ How can I show that $$\langle x, Ax \rangle + \langle b, x \rangle = c$$ can be transformed by ...
0
votes
0answers
29 views

Transformation of a sum of dot products

I'm not quite sure about this, So I'd like if someone could help me. Can someone explain to me how they get from 2 to 3? \begin{align}\left<2u,u+v\right> &= 0\tag1\\ ...
0
votes
1answer
36 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: $$\langle x,y\rangle = \langle[x]_{B} , [y]_B \rangle{st}$$ ...
2
votes
1answer
69 views

Why is the Cauchy-Schwarz inequality considered to be so important?

I've read in the book "Linear Algebra done right" by Axler that the Cauchy-Schwarz inequality is one of the most important results in mathematics. However, in what the book covers and what we have ...
1
vote
2answers
27 views

Given a symmetric matrix A, find an orthogonal matrix S such that $S^TAS$ is a diagonal matrix

Given the symmetric matrix: $$A = \left( \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} \right)$$ find an orthogonal matrix $S$ such that $S^TAS$ is a ...
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 $Q$
0
votes
1answer
38 views

Linear algebra: determining if something is an inner product space

If I have a potential inner product space over P2, where $\left< p, q\right> = p(0)q(0)$ How do I determine whether or not it is an inner product space? Using the four axioms I have: ...
3
votes
1answer
50 views

Determinant (or positive definiteness) of a Hankel matrix

I need to prove that the Hankel matrix given by $a_{ij}=\frac{1}{i+j}$ is positive definite. It turns out that it is a special case of the Cauchy matrices, and the determinant is given by the Cauchy ...
0
votes
2answers
25 views

Proving the image of inner product map is whole subspace

I'm doing a specimen exam question and they often have typos and missed pieces of necessary information. I think the question I'm doing might be one such example, but am not sure: We're given that ...
6
votes
2answers
42 views

Show that $V = U^\perp \bigoplus U$

If $(V,\langle , \rangle)$ is a Euclidean vector space, $U \subseteq V$ is a subspace of V and $U^\perp := \{v \in V | \langle v,u \rangle = 0, \forall u \in U\}$. Show $V = U^\perp \bigoplus U$ In ...
0
votes
1answer
51 views

Orthogonal projection in Inner product space

Let V be $n$-Dimensional ($n\ge1$) inner product space . Let $T:V \rightarrow V$ be a linear map which maintains $ T^2=T$ , $\forall v \in V\ ||Tv||\le||v||$. Prove that there is exists a subspace ...
0
votes
1answer
77 views

Symmetric matrix over inner product space

I try really hard to prove this Question. let $A_{nXn}(\mathbb{R})$ Symmetric matrix $A=A^t$ let $\lambda$ be the greatest Eigenvalue of A. we will define over the field $\mathbb{R}$ with the ...
-1
votes
3answers
63 views

Consider a set $S$ of unit vectors in $\mathbb R^2$ such that $\left<x,y\right>=-\frac12$ if $x,y\in S,x\ne y$.

This is a question from an entrance exam paper. Consider a set $S$ of unit vectors in $\mathbb R^2$ such that $\left<x,y\right>=-\frac12$ if $x,y\in S,x\ne y$.Then it is necessarily true that ...
1
vote
1answer
38 views

Relationship between matrix 2-norm and orthogonal basis of eigenvectors

Given the following matrix: $$ A = \left( \begin{array}{cc} 3 & 4 \\ 0 & 5 \\ \end{array} \right)$$ calculate $\|A\|_2$, with $\|A\|_2 = max_{x \in \mathbb{R}^2 -\{0\}} \frac{\langle Ax,Ax ...
2
votes
2answers
24 views

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 ...
1
vote
1answer
30 views

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 ...
4
votes
1answer
33 views

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 ...
0
votes
0answers
18 views

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 ...
1
vote
1answer
18 views

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 ...
3
votes
1answer
39 views

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 ...
0
votes
1answer
40 views

Isomorphism of inner product spaces

I want to use this in a proof, however I don't know how to prove it itself. I feel as though it's easy to prove by definition but I'm not quite sure.. A linear map V→W between two finite dimensional ...
2
votes
2answers
60 views

Inner product and linear transformation

Let V be an inner product space over $\mathbb{R}$ with inner product ⟨ , ⟩. Let $L:V\rightarrow\mathbb{R}$ be a linear transformation. Show that there is a $\vec{u}\in{V}$ such that $L(\vec{x}) = ...
0
votes
1answer
37 views

$A \in M_2(\mathbb{R}^2)$,$\langle x,y \rangle = x^TAy$ is an inner product iff $\alpha > 0, det(A) > 0$

Show that given $A=\left( \begin{array}{cc}\alpha &\beta\\\beta&\delta\\\end{array}\right) \in M_2(\mathbb{R}^2)$, $\langle x,y \rangle = x^TAy, (x,y \in \mathbb{R}^2)$, defines an inner ...