Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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complex problem in linear algebra

Let $A$ be an $n$ by $n$ matrix. Let $D$ be an $n$ by $n$ diagonal matrix with distinct diagonal entries, and let $u$ be an $n$ by $1$ column vector with all non-zero entries. Let $Aq=\lambda q$ with ...
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1answer
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Can a low-rank matrix set have nonempty interior?

The answer to this question may be super simple, but it is very not obvious to me. Consider the space $S^n$ of symmetric $n\times n$ matrices. Consider $T\subset S$ the set of rank $n-1$ matrices. ...
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full row rank matrix and 2-norm solution

Let $A$ be an $m$ by $n$ matrix with $m < n$ and with rank($A$)=$m$. Consider the system $Ax=b.$ (i) Find a particular solution to the row space of $A$. (ii)Find the projection onto the row space ...
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1answer
34 views

Let $A$, $B$ be two $3\times3$ commuting matrices, where $A$ is nilpotent and $\operatorname{tr}B = 0$. Prove that $ABA = O$

Let $A$ and $B$ be two $3\times3$ commuting matrices, where $A$ is nilpotent and $\operatorname{tr}B = 0$. Prove that $ABA = 0$. Progress I know that $ABA=0 \implies A^2B=0$. Here ...
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2answers
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What is the geometric interpretation of a vector squared?

I'm working through Introduction to Space Dynamics by William Tyrrell Thomson. I am having to do a lot of research to make it through even small parts, but I am unable to find information to make me ...
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Request for clarification about linear combinations

I need help understanding the basis of this statement in Axler's Linear Algebra Done Right, found on page 86 of the second edition: Because ($\vec{v_{1}}, \ldots, \vec{v_{n}}$) is a basis of $V$, we ...
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1answer
28 views

Show the inverse of a bijective function is bijective

We have a function $\varphi:G\rightarrow H$ is an isomorphism, show its inverse $\varphi^{-1}:H\rightarrow G$ is also an isomorphism I am fine with showing it to be a homomorphism and surjective, ...
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1answer
25 views

Change of basis matrix - part of a proof

I'm trying to understand a proof from Comprehensive Introduction to Linear Algebra (page 244) I can't really figure out what steps have been taken to get from eq. 1. to eq. 2. It's just ...
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1answer
15 views

Given $n$ linear functionals $f_k(x_1,\dotsc,x_n) = \sum_{j=1}^n (k-j)x_j$, what is the dimension of the subspace they annihilate?

Let $F$ be a subfield of the complex numbers. We define $n$ linear functionals on $F^n$ ($n \geq 2$) by $f_k(x_1, \dotsc, x_n) = \sum_{j=1}^n (k-j) x_j$, $1 \leq k \leq n$. What is the dimension of ...
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2answers
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To determine Rank of Linear Transformation

Question is to find the rank of $T_1 $and $T_2$ Since the composition is bijective so rank of $T_1T_2 = m$. But how do I get the ranks of$ T_1 $and$ T_2 $from here? Thanks.
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1answer
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Set of linear equations with coefficients - solution using matrices

I have a set of linear equations: \begin{matrix} ax_{1}& {}+bx_{2}& {}+x_{3}& & =0\\ cx_{1}& {}+dx_{2}& &{}-x_{4} & =0\\ & {}-ex_{2}& ...
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1answer
52 views

Help Understanding Proof from Linear Algebra Done Right

I'm doing a self-study of Axler's Linear Algebra Done Right, and am looking for some help understanding a step in the proof of Proposition 5.21, appearing on page 89 of the second edition. An ...
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0answers
33 views

A matrix transformation from R^4 to R^3 - linear algebra - how to find the image of a point

I'm trying to revise for an upcoming exam on linear algebra and have come across this question. I do not understand the line "the image of a point (x1, x2, x3, x4) can be computed from the defining ...
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Algorithm for vector space transformation [on hold]

In my text book I've got an example which is as follows: Create an algorithm which calculates coordinates of a point after a space transformation took place. Transformations may be scaling or ...
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36 views

General form for the rotation of a function.

When rotating linear functions, I would approach the task geometrically (find invariant point etc.), yet I tried using a matrix which worked nicely. This was what I did to rotate $y=2x+1$ by ...
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3answers
37 views

Is $ x^n-y^n$ is a product of coprime factors?

In the expression: $x^n-y^n$, if $n>2$ and $x,y$ are relatively prime, are the factors $x-y$ and $ x^{n-1}+x^{n-2}y+.....$ always coprime? Why? Please exclude the cases where $x-y=\pm 1$ and $\pm ...
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1answer
56 views

How to solve this kind of problem?

I've just found the following problem: $\quad\quad$ $\quad\quad$ $\quad\,$ And I believe that it could be done with something in combinatorics, my feeling is that generating functions would ...
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1answer
48 views

How do i find eigen vector

I need to find corresponding eign vector forthis problem Any hints for this .Thanks
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1answer
46 views

Vector Spaces: Tensor Product

Reference Foundation for: Hilbert Spaces: Tensor Product Problem Given a vector spaces $V$ and $W$. Take its algebraic tensor product: $\tau:V\times W\to V\otimes W$ How to prove that the image ...
2
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3answers
134 views

Linear dependency of nilpotent matrices

I would like to prove that four $2\times 2$ nilpotent matrices are always linearly dependent, using the Cayley-Hamilton theorem or the minimal polynomial in some way. I think I have proved the ...
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7answers
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Given matrix P such that $P^{102 } =0 $ , to show that $P^{2} = 0$.

P is given to be a 2×2 matrix such that $P^{102} = 0$. How to show that $P^{2} =0 $?
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1answer
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If A and A' are approximately the same, are their principal components/SVD very close?

If we have that two matrices $A\approx A'$ within some guaranteed error bound for each term, and $A=U\Sigma V$ is the singular value decomposition for $A$, and $A'=U'\Sigma' V'$ is the SVD for $A'$, ...
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3answers
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To find the two dimensional subspace of $R^{3}$

I am stuck with this question .Kindly help me to get through this Option A is of 1 dimension so it cannot be answer but all other options are looking fine to me , What i am missing ? THANKS
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2answers
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Using inverse of matrix A as approximate inverse of matrix that is very close to A

Say we have two matrices, $A$ and $A'$ so that $A\approx A'$, and we have the inverse of $A$, $B$, where $AB=I$, and the inverse of $A'$ where $A'B'=I$. If we have some guarantee about how big any ...
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Proving matrix similarity for given matrices

Let $\mathbb{F}$ be a field, and let $A=(a_{ij})_{i,j=1}^n$ and $B=(b_{ij})_{i,j=1}^n$ be matrices in $M_n(\mathbb{F})$ such that: a. $b_{ij}=0 \iff a_{ij}=0$ $ \forall 1 \leq i,j \leq n$ b. ...
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1answer
15 views

Null/Col/Row space be a line\plane through the origin?

For a $4\times3$ matrix can the nullspace, the column space and row space all be lines through the origin? For a $2\times4$ matrix can the nullspace, the column space and row space all be planes ...
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1answer
23 views

Two square matrices with the same minimial polynomial are similar for $n=5$ or $n=6$

Let $\mathbb{F}$ be a field, $\lambda \in \mathbb{F}$ and $A,B \in M_n(\mathbb{F})$ such that $m_A(x)=m_B(x)=(x-\lambda)^k$ and such that the geometric multiplicity of $\lambda$ in $A$ equals to the ...
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0answers
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How can i find column of matrix corresponds to row of matrix's inverse

let $Y=X\beta$ be an equation of matrix and let $X$ be an invertible $n\times n$ matrix, $Y$ be $n \times 1$ matrix, $\beta$ be $n \times 1$ matrix. $$\begin{bmatrix} y_1 \\ y_2 \\y_3 ...
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38 views

What does adjoint of a linear map?

I have been studying Linear Algebra from Axler, and I came across adjoint of a linear map. I understood the properties and concept of adjoint, basically $\langle Tv,w \rangle = \langle v,T^*w \rangle ...
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1answer
36 views

conditions for $A +B$ to be semi-definite.

Suppose $A$ is a positive definite real matrix, and $B$ is symmetric and real matrix with $B_{ii}>0$. Are there conditions on $\sup_{j}|B_{ij}|$ that can guarantee $A+B$ is semi-definite. ...
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a question regarding wronskian

I was working on following problem: Let $y_1$ and $y_2$ be solutions of $x^2y'' + y' + (\sin x)y = 0$ satisfying $y_1(0) = 0, y_1'(0)=1,y_2(0) = 1, y_2'(0)=0 $. I worked like following: since ...
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Does the method of Alternating Projections provide a link between the result for subspaces and that for a hyper plane?

We have the results for the projection onto a subspace $V$ in $\mathbb{R}^n$, for example, the projection of any $x\in \mathbb{R}^n$ onto $V$ is defined by the matrix characterization ...
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2answers
41 views

Nullspace, row space, column space in $m\times n$ matrices [on hold]

For a $4\times 3$ matrix can the nullspace, the column space and row space all be a line through the origin? For a $2\times 4$ matrix can the nullspace, the column space and row space all be a plane ...
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1answer
32 views

What can we say about output of Gram–Schmidt process

Given $\{x_1, \dots, x_{n-1}\}$ linearly independent vectors and $x_n \in \operatorname{span}\{x_1, \dots, x_{n-1}\}$ and let $\{\hat{x_1}, \dots, \hat{x_{n-1}}, \hat{{x_n}}\}$ be the output of the ...
3
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1answer
57 views

Does every isomorphism between $V$ and $V^*$ send some basis to its dual basis?

Suppose that I have a vector space isomorphism $\theta: V \to V^*$ where $V$ is any vector space (probably over $\mathbb{C}$ is required) and $V^*$ is its dual space. Is it always possible to find a ...
4
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2answers
40 views

Eigenvalues of hermitian plus skew-hermitian PSD matrix

I was wondering, suppose you have a matrix of the form $A=B+iCC^\dagger$ where $^\dagger$ denotes the hermitian conjugate. $B$ is hermitian and $CC^\dagger$ is obviously hermitian positive ...
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2answers
26 views

Orthographic projection of point $[0, 0, 0]$

What is the easiest way to calculate orthographic projection of point $[0, 0, 0]$ on a plane given by formula $x - y + z = 1$?
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3answers
276 views

Every invertible linear transformation can be perturbed a bit without destroying invertbility, Neumann series

Let $T: V \to V$ be any linear transformation on a real or complex vector space $V$. Show that there exists $\epsilon_0 > 0$ $($depending on $T$$)$ so that $I + \epsilon T$ is invertible for any ...
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1answer
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How can a $k\times (k-m)$ matrix be multiplied by a $k\times m$ matrix?

While reading a book on differential geometry, I came across this line: Since the differential $d\psi_0(x_0):\mathbb R^m\to \mathbb R^k$ is injective, there is a matrix $B\in \mathbb R^{k\times ...
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1answer
30 views

Finding basis of inverse image

Let $\psi $ be a linear transformation such that$$\psi ([x_1,x_2,x_3,x_4])=[x_1+x_3+x_4, -x_2-x_4,x_1+x_2+x_3+2x_4].$$ Find basis of inverse image $\psi^{-1}(W)$ of subspace ...
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Is $\frac{F(b)-f(a)}{b-a}x < f(x)$ for $x\in[a,b]$? [on hold]

As stated in the description, I want to know whether the following statement is true or not Is $\frac{F(b)-f(a)}{b-a}x < f(x)$ for $x\in[a,b]$?
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2answers
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Why is this map not surjective at the origin?

$f:\mathbb{R}^{n+1} \rightarrow \mathbb{R}$ $f(x) = |x|^2$ Then the derivative map is $df_{x}(v)=2\sum_{i}{x^iv^i}$ is surjective except at 0. Is it because at 0 df only goes to 0, and doesn't ...
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1answer
30 views

Find basis of an image

Let $ \psi : \mathbb{R}^4 \rightarrow \mathbb{R}^3$ be a linear transformation described by a formula $$\psi ([x_1,x_2,x_3,x_4])=[x_1+x_3+x_4, -x_2-x_4,x_1+x_2+x_3+2x_4].$$ Find basis of image ...
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Isomorphism type of a finite matrix group

Give a known group to which $SO_5$ is isomorphic. Where $SO_5$ consists of $$\begin{pmatrix} \cos\frac{2k\pi}{5} & \sin\frac{2k\pi}{5} \\ -\sin\frac{2k\pi}{5} & \cos\frac{2k\pi}{5} ...
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Physical or geometric meaning of the trace of a matrix

The geometric meaning of the determinant of a matrix as an area or a volume is dealt with in many textbooks. However, I don't know if the trace of a matrix has a geometric meaning too. Is there ...
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Direct sum of two spaces

Let $\alpha_1=[1,1,0,1]$, $\alpha_2=[1,0,1,1], \alpha_3=[1,1,1,1],\alpha_4=[0,1,1,1]$ be a vectors from $\mathbb{R}^4$ let $U=span(\alpha_1, \alpha_2) \ and \ W=span(\alpha_3, \alpha_4)$ Check that ...
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1answer
87 views

Determinant: Continuity

Reference Build-up on: Determinant: Definition Problem Given a vector space $V$. Consider an endomorphism $T:V\to V$. Define its determinant $\det:\mathcal{L}(V)\to\mathbb{C}$. Introduce a norm ...
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41 views

Where can Gaussian Elimination be used?

I have searched for this and came to know about it that it is traditionally used to solve linear equations, finding determinant, rank of matrix, inverse of matrix. There was a problem on codechef: ...
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1answer
15 views

a question about general and particular solutions

We have a $3\times 6$ matrix $A$ with rank $3$ (this is all the information we have, no matrix given). Here comes the questions: What is the number of free variables in the solution to the system ...
2
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1answer
31 views

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,.,1,0,.)$ and a bounded linear functional $\Phi$ find $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges?

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,...,1,0,...)$ and a bounded linear functional $\Phi$ find a value of $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges for $b_n=\Phi(e_n)$? Ok so since ...