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
25 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
24 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
28 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
48 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
52 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
38 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
83 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
33 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
41 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
38 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
51 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
43 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
79 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
59 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
55 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
44 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
52 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
62 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
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: $$\langle x,y\rangle = \langle[x]_{B} , [y]_B \rangle{st}$$ ...
2
votes
1answer
67 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
24 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
37 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
44 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 ...
5
votes
2answers
39 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
50 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
76 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
61 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
30 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
32 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
38 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
37 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
36 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 ...
2
votes
1answer
36 views

Inner product definition, definite positive

I'm reading Hoffman and Kunze's linear algebra book and I'm a bit stuck with it's definition of inner product. One of the properties of the inner product is said to be that the product of any vector ...
6
votes
0answers
76 views

Inner product on $C(\mathbb R)$

With Axiom of choice it is possible to construct an inner product on $C(\mathbb R)$. Of course, the space would be not complete under the norm induced by the inner product. My question is, is it ...
1
vote
1answer
86 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
160 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 = ...
1
vote
3answers
62 views

Inner Product Spaces, suggestion for book.

Can you suggest me name of some books which would help me visualize IPS better? Like, books having diagrams and stuff?
1
vote
3answers
56 views

Difference between Euclidean space and inner product space?

Is it that Inner product space can have infinite dimensions?