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 ...

learn more… | top users | synonyms

1
vote
2answers
25 views

What row-operations allow this $\operatorname{Mat}_{2\times2} (\mathbb{R})$

$$ A = \begin{pmatrix} 1 & r \\ s & 1 \\ \end{pmatrix} \Rightarrow \begin{pmatrix} 1 & r \\ 0 & 1-s \cdot r \\ \end{pmatrix} = B \quad\quad r,s \in \mathbb{R} $$ Matrix B is ...
0
votes
1answer
21 views

Hermitian Matrix Inequality

If we have {$A_{ij}\}_{n*n}$ a Hermitian matrix. v=($v_1,v_2..v_n$), w=($w_1,w_2...w_n$) are two complex vectors. Then how can I show the inequality |$\sum_{i,j=1}^nA_{ij}v_i\overline{w_j}$|$\leq ...
12
votes
1answer
339 views

Convexity of Matrix Exponential

Consider the function $A: \mathbb{R}^n \rightarrow \mathbb{R}^{n \times n}$ defined as $$ A( x ) := \left[ \begin{matrix} x_1 & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & x_2 & ...
0
votes
0answers
24 views

Subspaces of vector space and their spans

$\def\sp{\operatorname{sp}}$ Given 2 subspaces of V, and $T\subseteq \sp(K)$ and $K\subseteq \sp(T)$ then $\sp(K)=\sp(T)$? If $ T\subseteq \sp(T)\subseteq \sp(K)$ and $K\subseteq ...
1
vote
1answer
55 views

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}& ...
0
votes
2answers
35 views

Proof for pythagoras theorem

Let $f,g$ orthogonals to each other. $${\left\| {f + g} \right\|^2} = \left<f,f\right>+\left<g,f\right>+\left<f,g\right>+\left<g,g\right> = {\left\| f \right\|^2} + {\left\| g ...
1
vote
2answers
67 views

Derivative of a trace w.r.t matrix within log of matrix sums

I'm trying to solve an optimization (sub)problem and am running into trouble with a tricky derivative. I'd like to find the matrix $C \in \mathbb{R}^{n\times d}_+$ which minimizes the following ...
1
vote
1answer
73 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 ...
2
votes
1answer
23 views

Find a hyperplane not intersecting $S$

I am struggling with the following problem: Let $K$ be an infinite field, $V$ an $n$-dimensional $K$-vector space, $S \subset V$ a finite subset with $0 \notin S$. Prove that there exists a subspace ...
1
vote
2answers
46 views

Jordan form of a matrix

Let $$A = \left( {\matrix{ 0 & 1 & 0 & 0 \cr 0 & 0 & 2 & 0 \cr 0 & 0 & 0 & 3 \cr 0 & 0 & 0 & 0 \cr } } \right)$$ The ...
2
votes
3answers
31 views

What ring-sum of vector spaces can possibly mean?

I'm given this test assignment, and I can't decipher what it says. Would you kindly help me? Here's the assignment itself: Let $U$ and $W$ be sub-spaces of the linear vector space $V$ s.t. $U ...
0
votes
1answer
26 views

How to find the velocity and accelaration in a 3d space with 6 degrees of freedom?

I have the following rigid body: I assume that the body is a symmetric cylinder.x,y,z are the axes of the reference frame resulting from a transformation involving three orthogonal rotations ...
0
votes
1answer
27 views

Proving $\mathbf x$ is a $n\times 1$ vector and $\mathbf A$ an $n\times n$ matrix, then $\mathbf x'\mathbf A \mathbf x = \text{tr} (\mathbf {Axx}')$

If $\mathbf x$ is a $n\times 1$ vector and $\mathbf A$ an $n\times n$ matrix, then $\mathbf x'\mathbf A \mathbf x = \text{tr} (\mathbf {Axx}') (\mathbf A'=transpose A) $
1
vote
1answer
61 views

Book comparison, Linear Algebra

so next semester (Spring 2015) I'm taking a Linear Algebra class. I was wondering if anyone who's had this book "Linear Algebra and Its Applications, 4th Edition - by David C. Lay" can give me an ...
2
votes
2answers
28 views

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. ...
72
votes
0answers
2k views

How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrices $A,B,C\in M_{n}(C)$ be Hermitian and Positive definite matrices, such that:$$A+B+C=I_{n}$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ ...
0
votes
1answer
23 views

Linear transformation to higher dimensional space.

There is a 7-by-6 matrix $H$ given. Its rank is 6. I'd like to design a 6-by-5 matrix $D$ such that the following holds: $ \left[ \begin{array}{l} l_1(a_1, a_2, a_3, a_4) \\ l_2(a_1, a_2, a_3, a_4) ...
1
vote
2answers
331 views

Finding the inner product generated by a matrix

In each part, use the given inner product on $R^2$ to find $\|\vec w\|$, where $\vec w\ = (-1, 3)$. Then the problem lists different inner products to use to find the norm but the one I'm ...
2
votes
0answers
40 views

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 ...
2
votes
7answers
573 views

How to solve these equations for x and y..

equations are $(x-y)(x+2y)(2x+y) = 20$ and $x^2+xy+y^2 = 7$ i want the METHOD not the solutions
6
votes
2answers
303 views

Origin of the modern definition of the tensor product

Due to whom is the modern (i.e. via its universal property) definition of the tensor product, and in which article was it communicated?
1
vote
0answers
35 views

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 ...
1
vote
3answers
39 views

orthogonal and special orthogonal group of dimension $2$, group of isometries of $S_1$, $\mathbb{R}^2$ [closed]

In my abstract algebra class, my teacher gave us this problem as to help review for the final. Unfortunately, I am not very well versed with linear algebra so I don't understand all that well what ...
8
votes
1answer
32 views

ordered partition, block matrix given by $r_j \times r_j$ nilpotent Jordan blocks is nilpotent, rational canonical form, jordan canonical form

Let $F$ be a field. For an integer $n \ge 1$, and ordered partition of $n$ is a sequence $\underline{r} = \{r_1, \dots, r_m\}$ of positive integers such that $r_1 \le \dots \le r_m$ and $\sum r_j = ...
1
vote
1answer
487 views

Total unimodularity of matrix with consecutive ones property

A matrix has the consecutive ones property (often abbreviated C1P) if its every row (or column, for column-oriented C1P) is of the form $(0,\ldots,0,1,\ldots,1,0,\ldots,0)$. There is a theorem which ...
12
votes
4answers
923 views

Difference between Ring and Algebra?

In mathematics, I want to know what is indeed the difference between a ring and an algebra? Thanks!
1
vote
1answer
58 views

About diagonalization

"Let A = $\begin{bmatrix}1 & 1 & 4\\0 & 3 & -4\\0&0&-1\end{bmatrix}$. Is the matrix A diagonalizable? If so find a matrix P that diagonalizes A. Can you write A as a linear ...
2
votes
2answers
81 views

Show that this mapping is a linear transformation [on hold]

Show that the map v = [v]E → [v]B for all v ∈ Rn defines a linear transformation TB : Rn → Rn. B = {b1, b2, b3,...,bn} is a basis of Rn. Any vector v ∈ Rn can be uniquely expressed as a linear ...
2
votes
1answer
19 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 ...
3
votes
1answer
37 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 ...
1
vote
2answers
25 views

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.
3
votes
1answer
64 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 ...
0
votes
0answers
39 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 ...
-1
votes
0answers
23 views

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 ...
1
vote
1answer
60 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 ...
2
votes
3answers
43 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 ...
-3
votes
1answer
54 views

How do i find eigen vector

I need to find corresponding eign vector forthis problem Any hints for this .Thanks
1
vote
1answer
57 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 ...
0
votes
1answer
3k views

Inverse of upper triangular matrix

I have an upper triangular matrix that I want to solve the inverse for. I have $[Ax_i e_i]$ where $x_i$ is the $i$th column from the inverse of $A$ and $e_i$ is the $i$th column of the identity ...
1
vote
2answers
54 views

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 ...
1
vote
1answer
90 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 ...
0
votes
7answers
118 views

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 $?
0
votes
1answer
17 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 ...
1
vote
1answer
134 views

Gram Schmidt Process on complex space

Let $\mathbb{C}^3$ be equipped with the standard complex inner product. Apply the Gram-Schmidt process to the basis: $v_1=(1,0,i)^t$, $v_2=(-1,i,1)^t$, $v_3=(0,-1,i+1)^t$ to find an orthonormal ...
2
votes
3answers
168 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 ...
2
votes
1answer
28 views

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 $A'$, ...
0
votes
3answers
17 views

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
1
vote
1answer
29 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 ...
0
votes
0answers
35 views

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. ...
2
votes
0answers
40 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 ...