Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Addition of Kronecker Product Matrices

Summary: Is it possible to write $A_1 \otimes A_2 + B_1 \otimes B_2$ as some object which has nice properties again, preferably as a Kronecker product itself? Each of the matrices $A_i$, $B_i$ can ...
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votes
1answer
39 views

True or false? (linear algebra) [on hold]

If $u$ and $v$ are two solutions of $Ax$ = $0$, then any vector in $Span(u, v)$ is also a solution of $Ax = 0$. I have doubts with $span(u,v)$. I do not know if the same thing $span(u\cup v)$ or ...
0
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1answer
461 views

Change of (orthonormal) basis.

As I see it, the author says that $[Tv]_{e} = A[v]_{e}$ in the last paragraph. How do I see that ? I think I've jusitied the first entry in $[Tv]_{e} = A[v]_{e}$ as follows \begin{align*} \langle ...
0
votes
1answer
21 views

$Y$ coordinate of a point that lies on a line [on hold]

Given two points $A$ and $B$, for example $A(1,5),\,B(15,2)$, what is the $y$ coordinate of a point $C(x,y)$ lying on the straight line $AB$?
6
votes
0answers
57 views

In a finite dimensional inner product space with $T ∈ L(V)$, show that $\langle u,v\rangle = \langle T(u),T(v)\rangle$ implies $T$ is invertible.

Here is how I've tried to go about it, and I'm curious if it's true or if I'm way off base. T is invertible iff null$(T)=\{0\}$. Let $v∈V$ and suppose $T(v)=0$. If we can show that $v=0$, then $T$ is ...
0
votes
0answers
22 views

bilinear form and positive definiteness

Let $B$ a symmetric bilinear form on an $n$ dimensional vector space $E$ with signature $(n-1,1)$. Then there exists a hyperplane $H$ in $E$ in which $B$ is positive definite. How to prove this? Is ...
3
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0answers
94 views

Problem involving subspaces and linear transformations

I'm asking for some opinions about my proof! $V$ and $W$ are vector spaces, and $T : V \rightarrow W$ is a linear transformation. $Z$ is a subspace of $W$, and $U$ is the set of all $\textbf{x} \in ...
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3answers
7k views

Prove rank $A^TA$ = rank $A$ for any $A_{m \times n}$

How can I prove rank $A^TA$ = rank $A$ for any $A_{m \times n}$? This is an exercise in my textbook associated with orthogonal projections and Gram–Schmidt process but I am unsure how they are ...
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vote
2answers
24 views

Norm of diference of matrices of different rank

Suppose $A$ is a $n\times n$ matrix of rank $k$ that has Euclidean norm equal to $1$. Given $p<k$, and $\epsilon>0$, can we always find a norm one matrix $B$ of rank $p$ such that ...
3
votes
3answers
27 views

multiplication of finite sum (inner product space)

I am having difficulty to understand the first line of the proof of theorem 3.22 below. (taken from a linear analysis book) Why need to be different index, i.e. $m,n$ when multiplying the two sums? ...
2
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1answer
26 views

Prove a semi-positive operator $T$ is an isometry if and only if $T$ is the identity operator.

Prove a semi-positive operator $T$ is an isometry if and only if $T$ is the identity operator. I was thinking that semi-positive means if $T$ is self-adjoint ($T^{*}=T$) and $\langle T(u),u\rangle ...
0
votes
1answer
22 views

representation of a map with respect to 2 bases

From Heffron, p.231 Consider the two linear functions $h:$ ${R}^3$ $\longrightarrow$ $\mathcal{P}_2$ and ${g}: \mathcal{P}_2 → M_{2x2}$ $ \left( \begin{array}{ccc} a \\ b \\ c \end{array} ...
3
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0answers
20 views

Compact-open topology on $\operatorname{Hom}_\mathbb{R}(V,W)$

Suppose $V$ and $W$ are finite-dimensional real vector spaces, and I give $\operatorname{Hom}_{\mathbb{R}}(V,W) \cong V^* \otimes W$ its usual vector space topology. Does this agree with the subspace ...
0
votes
1answer
19 views

Systems generators that are not linearly independent

Good evening, I would find sets that are generators of vector spaces, but they are not linearly independent, ie they are generating space but are not a basis for it. For example for these spaces: ...
2
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0answers
11 views

Where can I learn properties about spaces of linearly independent projectors?

I am interested in characterizing the space of all collections of $d^2$ linearly independent projectors on the Hilbert space $\mathbb{C}^d$. The linear independence I desire is in the vector space of ...
0
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0answers
24 views

Let $V = R^3$ and let $U$ be the subspace spanned by $A= \{(-3,-2,0),(4,-1,2)\}$. Is there a subspace $W$ of $V$ such that the following holds?

Let $V = \mathbb R^3$ and let $U$ be the subspace spanned by $A= \{(-3,-2,0),(4,-1,2)\}$. Is there a subspace $W$ of $V$ such that the following holds? $$W \nsubseteq U$$ ...
2
votes
1answer
531 views

How come that HSL can contain more information than RGB?

I have noticed weird thing when working with HSL - unlike RGB, it has some blind spots where certain value just does not matter. I'm sure we were taught about this when I had Linear algebra lectures ...
0
votes
1answer
11 views

Condition number of preconditioned system

Suppose we are solving an ill-conditioned system $Ax = b$, and we are trying to solve it using preconditioned technique. Given $\kappa (T)\approx \kappa(A)$, where $\kappa(A)$ is condition number of ...
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0answers
52 views

Cramer's rule doesn't work here?

I tried to solve the following system: $$A_2\cdot 2\mathrm{i}\sin( \beta a) = B_3\exp(- \alpha a)$$ $$\mathrm{i} \beta A_2 2\cos( \beta a) = - \alpha B_3\exp(- \alpha a)$$ Then I got $A_2=0 ...
3
votes
2answers
60 views

Efficiently solving many sets of linear equations without inversion or factorization

Suppose I have the normal set of linear equations $Ax = b$. If I can store and manipulate $A$ I have a variety of techniques available to me such as inversion, factorization, or an iterative method. ...
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4answers
43 views

How do I find orthonormal basis of $U$?

Let $U$ be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. How do I find orthonormal basis of $U$?
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How do I set the vector $\vec{v} = (0, 0, 1, 0, 0)^T$ as $\vec{\vec{v}}=\vec{u}+\vec{w}$ with $\vec{u}\in U$ and $ \vec{w}\in U^\perp$? [on hold]

Let U be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. enter image description here
3
votes
2answers
58 views

Prove that $\|u - v\| \ge \|u\| - \|v\|$ for any $u, v \in \mathbb R^n$

Checking to see if the argument below works. Consider $\|u - v\|^2 = \|u\|^2 - 2(u \cdot v) + \|v\|^2$, by Cauchy-Schwarz: $\|u\|^2 - 2(u \cdot v) +\|v\|^2 \ge \|u\|^2 - 2(\|u\| \|v\|) + \|v\|^2 = ...
0
votes
1answer
35 views

Vector norm of $\mathbb R^n$, why is $p$-norm$\leq q$-norm if $p\geq q$?

Considering $$\infty\geq p\geq q\geq1$$ How can I show that the $p$-norm is smaller or equal to the $q$-norm? I can only show the case for $p=1, q=2$, but have no idea how to show others. Thank ...
0
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1answer
13 views

Generate a random binary full-rank rectangle matrix that is a basis of a subspace

Disclaimer: I think of vectors as row vectors. I have a full-rank $m \times n$ ($m < n$) binary matrix $B$ which is a basis of $m$-dimensional subspace $V \subset\mathbb F_2^n$ (i.e. subspace $V$ ...
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votes
2answers
20 views

How do I find a base of orthogonal complement $U^\perp$ of $U$ and determine the dimension of $U^\perp$? [on hold]

Let U be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. How do I find a base of orthogonal complement $U^\perp$ of $U$ and determine the dimension ...
4
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4answers
135 views

Finding a basis for the set of polynomials where f(1)=f(-1)=0

I have the vector space $V$ above that belongs to $\mathbb{F}$, and $V$ is the group of all polynomials that are of degree $3$. $W= \{ p \in V | p(1)=p(-1)=0\}$ 1.) Prove that W is a subspace of ...
1
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1answer
30 views

If $Ax = O$ has only one solutions, then the columns of A: ${v1, v2…,vn}$ span R?

I've been doing some excersices about inner product and I found something interesting but I don't know if my approach is correct at all. Supose that ${v_{1}, v_{2}, ..., v_{n}}$ is a base for a ...
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votes
2answers
24 views

The determinant of the transposing endomorphism

Let $K$ be a field and $f$ the endomorphism of $\mathcal M_n(K)$ that sends a matrix to its transpose. I want to determine the determinant of $f$. I know that since $f^2=id$ then $det(f)=1\ or \ -1 $ ...
0
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1answer
35 views

How do I determine $\min \left \| \vec{v}-\vec{u} \right \|_2$ for $\vec{u}\in U$?

Let $U=\lambda ((1, 0, 1, 0)^T,(1, 1, 0, 1)^T,(1, -1, 1; 0)^T$ is a subspace of $\mathbb{R}^4$. Determine for $\vec{v} = (1, 1, 1, 1)^T$ the vector $\vec{u}\in U$ minimal with $\left \| ...
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0answers
14 views

Can we exploit FFT for evaluating quadratic on gridded data with stationary covariance?

I would like to evaluate the quadratic $\mathbf{y}^{T}K^{-1}\mathbf{y}$ with the following assumptions: The entries of $\mathbf{y}$ are $y_i = f(\mathbf{x_i})$ which correspond to points on a ...
3
votes
0answers
126 views
+50

Spliting subspaces and finite fields

I'm sure that the following is true, but I can't prove it. Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a tower of finite fields and $A = \{\theta\in K: ...
2
votes
1answer
51 views

Basis for the intersection of two integer lattices

If $B_1$ and $B_2$ are the bases of two integer lattices $L_1$ and $L_2$, i.e. $L_1=\{B_1n:n\in\mathbb Z^d\}$ and $L_2=\{B_2n:n\in\mathbb Z^d\}$, is there an easy way to determine a basis for ...
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2answers
39 views

Is the set of periodic functions from $\mathbb{R}$ to $\mathbb{R}$ a subspace of $\mathbb{R}^\mathbb{R}$?

A function $f: \mathbb{R} \to \mathbb{R}$ is called periodic if there exists a positive number $p$ such that $f(x) = f(x + p)$ for all $x \in \mathbb{R}$. Is the set of periodic functions from ...
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votes
0answers
22 views

Suppose A and B are two matrices so that AB=0, which are true, if any, why? [on hold]

$ col(A)\subseteq null(B)$ $ null(A^TB^T) \subseteq null(A^T)$ $col(B^T) \perp col(A)$
0
votes
1answer
13 views

Finding $[T]_E$ for a basis that is composed of matrices

The linear transformation from $M_2 \rightarrow M_2 $ (Matrices that are 2x2) $T\left[\begin{pmatrix} a & b \\ 0 & d \\ \end{pmatrix}\right]=\begin{bmatrix} 1 ...
3
votes
1answer
55 views

Detecting singular system during Cholesky resolution

I am solving small linear systems with a symmetric positive matrix by the method of Cholesky, without pivoting. "Bad" matrices are detected when you take the square root of a diagonal element, which ...
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vote
2answers
18 views

Proving Rank of a matrix is greater than its sub matrix

How can I show that the rank of a matrix is always greater than or equal to the rank of every square matrix thereof.. I mean it is self evident to anyone who knows anything about rank of matrices but ...
3
votes
1answer
65 views

Result about Matrices of form $B(AB)^{-1}A$

I am trying to prove the following result. So far my only idea was to try using the formula for inversion of block matrices, but that did not get me very far. Any help will be much appreciated. ...
1
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3answers
27 views

Proving a basis in linear algebra

So at the moment I'm trying to go through proofs and I came across this one: Suppose $ P_n $ is the vector space of all polynomials with degree less than or equal to n. Prove that $ \{1, x − 1, ...
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0answers
21 views

Proving a subgroup is a basis for a space

Question: V (in R) is the subspace of all 2x2 Matrixes that are upper triangular. Prove that B is a basis for V. B= b1= $ \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ ...
3
votes
2answers
104 views

About Cauchy–Schwarz inequality

For the vectors $x$ and $y$, the Cauchy–Schwarz inequality reads $$ |x\cdot y|\leq||x||\cdot||y|| $$ Does this inequality only hold for 2-norm? Or for any norms? Thanks in advance.
8
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1answer
257 views
+100

Approximate spectral decomposition

See attempt below I am interested in effective and constructive computations for finding approximate spectral decompositions in some suitable format. Namely, let $A: H \rightarrow H$ be a Hermitian ...
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0answers
29 views

Change of Basis between linear Transformations

I am trying to get a better understanding in change of basis with matrices and linear transformations, therefore I am using several linear Transformations $^{i-1}A_i=\begin{bmatrix} \cos\theta_n ...
1
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1answer
18 views

Assume $T$ is a complex operator

Assume $T$ is a complex operator such that $T^{2}=T$. Prove that $Tr(T)$ is a non-negative integer. There is a remark in my book, Suppose the characteristic polynomial $\chi_{T}(x)$ factors intro ...
0
votes
4answers
46 views

What is a simple means of proving that 3 vectors belonging to $\Bbb{R}^2$ are linearly dependent?

For my linear algebra class, there is a 2 part problem that asks, given the set {(1 2), (-1 -1), (1 0)}, prove or disprove that it is linearly independent using the definition only AND then prove or ...
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2answers
177 views
4
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1answer
64 views

The tangent space of a vector space

I'm trying to show that there is a canonical isomorphism between a finite-dimensional vector space $V$ (regarded as a $C^\infty$ manifold) and its tangent space $T_vV, v\in V$, without using a basis, ...
16
votes
2answers
148 views

Is there a fundamental theorem of algebra for matrices?

The fundamental theorem of algebra says we can do this ($z\in\mathbb{C}$ of course) $$\sum_{k=0}^n a_kz^k= a_n\prod_{k=1}^n (z-\omega_k)=0$$ for some set $\{\omega_k \in\mathbb{C}\}_{k=1,2,\ldots , ...
1
vote
1answer
2k views

Condition number for non-square matrix?

From what I understand the condition number of a non-square matrix A is its largest singular value divided by its smallest nonzero singular value: $\kappa(A) = \sigma_1/\sigma_n $. Where ...