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 ...

learn more… | top users | synonyms (1)

1
vote
0answers
11 views

Realation between Matrices ?!

The answer to the following question could be trivial. Let $A_1, A_2$ be symmetric $n\times n$ matrices, $x=(x_1,\ldots,x_n)\in \mathbb{R}^n$. If the maximum is taken for over ($\|x\|=1,\, and ...
0
votes
1answer
18 views

Solution Set of System of ODE.

I am trying to find the solution of the system $$\begin{bmatrix}x_1\\x_2\end{bmatrix}'= \begin{bmatrix}1&3\\3&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}$$. I am given that ...
0
votes
1answer
20 views

Isomorphism matrix problem

So the question asks: Recall that $U^{2\times 2}$ is the vector space of 2X2 upper triangular matrices. Which of the following functions are isomorphisms? A. The function T: $U^{2\times 2}$ to ...
0
votes
1answer
14 views

How polynomials are represented in matrix form for Univariate Polynomial.

Represent this polynomial equation in matrix form $$P(x)=a_2 x^{2} +a_1x^{1} +a_0$$ ?
1
vote
0answers
35 views

Linear Algebra - Changing bases?

Let $T: \mathbb{R}^3\rightarrow M_{2\times2}(\mathbb{R})$ be the linear transformation defined by $T((a,b,c)) = \begin{bmatrix}a&5a\\c&3c\end{bmatrix}$. Consider the bases ...
0
votes
3answers
23 views

Explanation on derivation of this equation?

I'm extremely stuck on how my book was able to derive this equation. Basically, it says: Let $V = W = P_2(\mathbb{R})$. A basis for V is $1, 1+x, 1+x+x^2$. Define the linear transformation $T$ such ...
1
vote
1answer
20 views

Eigenvalues of matrix summation

Let $A$ be symmetric positive definite matrix with eigenvalues $\lambda_1,\lambda_2,\dots,\lambda_n$. Can we express the eigenvalues of $I-A$ using eigenvalues of $A$? I can't find properties of ...
0
votes
0answers
14 views

How to show that a set of elements is a basis for the ring of integers of a number field?

Let $K$ be a number field of degree $n$ (that is $[K:\mathbb{Q}]=n$) with ring of integers $\mathcal{O}_K$. I know that there exists algorithms to find $\mathcal{O}_K$ and hence determine a ...
0
votes
1answer
18 views

Spectrum Radius

Let $A$ be symmetric positive definite matrix and $\rho(A)$ denotes spectrum radius of $A$. How to prove that $\rho(I-\omega A)<1$ iff $0<\omega<2/\rho(A)$?
0
votes
1answer
30 views

Why are these functions linearly dependent?

$f1(x) = x$ $f2(x) = x^2$ $f3(x) = 5x - 4x^2$ From my understanding a set of functions are only linearly dependent if you can show that one function is simply a scaled version of another in the ...
0
votes
1answer
17 views

Linear Algebra Solving for Matrices

Consider the following matrix. $A$ = [ a b c ] [ d e f ] [ g h i ] Suppose that $\det(A) = −2$. Let $B$ be another $3 \times 3$ matrix (not ...
0
votes
2answers
13 views

Condition on matrix to ensure nontrivial Jordan canonical form

In my understanding, in order to ensure that a matrix $A$ has a nontrivial Jordan canonical form, one needs to come up with such a matrix whose geometric multiplicity is less than algebraic ...
0
votes
1answer
17 views

Proving Linear Independence Given Odd Absolute Values

With three vectors $a,b,c \in \mathbb{R}^3$, the magnitude of a$,b,c,a-b,b-c$, and $c-a$ are all odd integers (not necessarily distinct). How could you prove the three vectors are linearly ...
0
votes
0answers
19 views

Is this how to do matrix representation?

Say, $$f: \mathbb{Q}[t]_{4} \to \mathbb{Q}[t]_{4}$$ $f(q)=3q'''+2q''$ And we have the base $B=\{1,t,t^2,t^3,t^4\}$ and we wanted to find $[f]_{B}^{B}$ Then is this what would we do; $$f(1)=0$$ ...
1
vote
1answer
18 views

Stable eigenspace of $x'=Ax$

Given the system $\bf{x'}=Ax$, where $\bf{A}$$=\begin{bmatrix} -2 &0 &0 \\ 2& 1 & 0\\ 0 &0 &1 \end{bmatrix}$, the solution is $x(t) = \begin{bmatrix} e^{-2t} & 0 ...
0
votes
0answers
19 views

condition for having a positive solution to these linear equations.

Consider the following system of linear equations: $\sum_{j=1}^n c_{ij}\cdot x_{ij}=a_i$ for $j=1,\cdots,m$ and $\sum_{i=1}^m c_{ij}\cdot x_{ij}=b_j$ for $j=1,\cdots, n$ where for all $1\leq i\leq m$ ...
0
votes
0answers
17 views

$\bf{x'}=Ax$ with eigenvalues of multiplicity greater than $1$

Given the system $\bf{x'}=Ax$, where $\bf{A}$$=\begin{bmatrix} -2 &0 &0 \\ 2& 1 & 0\\ 0 &0 &1 \end{bmatrix}$, if I solve it by first finding matrix $\bf{P}$ and then ...
3
votes
3answers
48 views

Compute a natural number $n\geq 2$ s.t. $p\mid n \Longrightarrow p^2\nmid n$ AND $p-1\mid n \Longleftrightarrow p\mid n$ for all prime divisor p of n.

Question: Compute a natural number $n\geq 2$ that satisfies: For each prime divisor $p$ of $n$, $p^2$ does not divide $n$. For each prime number $p$, $p-1$ divides $n$ if and only if $p$ divides ...
2
votes
2answers
22 views

Basis of a Subspace given an Equation.

Hi, I was wondering if this question is asking us to find the basis of the kernel of this transformation from $\mathbb{R}^4 \rightarrow \mathbb{R}$ ? Thanks
2
votes
1answer
21 views

A linearly independent set that spans a space

So, in partial differential equations, we generate solutions for PDEs (kind of obviously). However, while the solutions we generate span the space of all solutions and are all linearly independent, ...
0
votes
0answers
19 views

How can I rearrange this formula to give it in terms of $t$? [on hold]

How can I rearrange the equation $$ e^{2t} = \frac{y^{2}(y+1)}{y-1} $$ to give it in the form $y = f(t)$?
0
votes
1answer
27 views

Linear maps (about rank and nullity)

$S:U\rightarrow V$ and $T:V\rightarrow W$ are linear maps. $U,V$ and $W$ are vector space over the same field. Prove: If $V=W$ and $T$ is non-singular then ...
2
votes
1answer
22 views

Distance between projections

Let $x,y,z \in \mathbb R^2$ such that $||x|| = ||y||= ||z|| = 1$. Project $z$ onto the lines defined by $x$ and $y$ as follows: \begin{equation} z_x = (z^\text{T}x) x, \ \ z_y = (z^\text{T}y) y, ...
0
votes
1answer
10 views

Subspace of $C^3$ that spanned by a set over C and over R

Given $A=$ $\left\{ {(1,2 + i,i),(1,3 + i,3 - i),(i,3i,4 + i)} \right\}$ Let $SP_CA$ be the linear space spanned by A over $C$ Let $SP_RA$ be the linear space spanned by A over $R$ what is the ...
1
vote
1answer
37 views

Is it possible to find determinant of a matrix by given the eigenvectors and the eigenvalues

If I already found the eigenvalues and eigenvectors of a particular matrix , is there an easy way to find the determinant of that matrix ?
0
votes
2answers
53 views

What is $\left | \left | A \right | \right |$ equals to in linear algebra?

Can someone please tell me what is this $\left | \left | A \right | \right |$ equals to? (determinant inside determinant)
0
votes
0answers
6 views

Bounding the perturbation between eigenvectors

Can somebody explain this part of the proof of a deduction from the Davis-Kahan $\sin \theta$ theorem? I understand how to get from: $||P_{u_1} - P_{v_1}|| \le \epsilon$ to $||P_{u_1}v_1 - v_1|| \le ...
2
votes
1answer
26 views

Linear Algebra - properties of positive semidefinite matrix

Let $A=(a_{ij})$ be a positive semi-definite, symmetric matrix, of order $3\times 3$ satisfying: $$ \Sigma_{j=1}^{3} a_{ij}=0 $$ for $i=1,2,3$ (i.e.- the sum of each row is zero). Prove: ...
0
votes
0answers
10 views

Bounding entries of random vector

Given a random vector $\mathbf{e} \in \mathbb{R}^n$, is it possible to count (or bound) the number of entries in $\mathbf{e}$ that have $|e_i| \ge 1/ \sqrt{n}$? It is known that entries in ...
0
votes
1answer
34 views

Linear Algebra. Is this question realte to combination and factorials?

I am not able to understand this question and what is the entries of matrix A exactly. Question Thanks.
0
votes
3answers
49 views

Linear vs. bilinear

I'm tripping over something elementary: Suppose $f:\mathbb{R^2}\rightarrow X$ is linear, then $f(x+y)=f(x)+f(y)$ for all vectors $x$ and $y$. Now suppose that $f$ is also bilinear and in particular ...
0
votes
1answer
33 views

$A$ is a linear transformation with eigenvalues

Let $A:\mathbb{R}^2\rightarrow\mathbb{R}^2$ be a linear transformation with eigenvalues $\frac{2}{3}$ amd $\frac{9}{5}$. Then, there exists a non zero vector $v\in\mathbb{R}^2$ such that ...
0
votes
0answers
16 views

$f: L \to L$ a diagonalizable operator with simple spectrum and $g: L \to L$ s.t $gf = fg$

I am making the exercises of Kostrikin and Manin (Linear Algebra and Geometry) and it has this question that I can't solve. Let $f: L \to L$ a diagonalizable operator with simple spectrum and $g: L ...
0
votes
1answer
20 views

Show linearity of this map, if and only if statement

Let $Γ_f = \{(s, t)\mid t = f(s)\} \subset S \times T$ Suppose that $U$ and $V$ are vector spaces and $f ∈ V^U$ (i.e. f is a map of underlying sets which is not necessarily linear). how do you show ...
0
votes
1answer
12 views

Uniqueness of the reduced echelon form - a doubt regarding the proof

Let's take a look at this proof It is claimed: It follows that both the $n$-th columns of $B$ and $C$ must contain leading 1's, for otherwise those columns would be free columns and we could ...
1
vote
2answers
77 views

What really is diagonalization?

If I have a square matrix $A$ representing a linear transformation $T:V\rightarrow V$ w.r.t the basis, $B=\{v_1,v_2..,v_n\}$ and $A$ is Hermitian. So we have $Av_n=\lambda_{n}v_n$ where ...
1
vote
0answers
14 views

Distance between two symmetric equations

I have been requested to solve this problem: Compute the distance between the lines: $L_{1}:\frac{x-2}{3}=\frac{y-5}{2}=\frac{z-1}{-1}$ and $L_{2}:\frac{x-4}{-4}=\frac{y-5}{4}=\frac{z+2}{1}$ This ...
3
votes
2answers
34 views

Invertible matrices modulo $29$. [duplicate]

Consider the $n\times n$ matrices with elements in $\mathbb{Z}_{29}$. How many of these are invertible? In total there are $29^{n^2}$ matrices of of dimension $n\times n$. Now I need to find how ...
1
vote
1answer
17 views

Identity regarding partial derivatives and polar representation

Let $f(x,y)$ be a differentiable function, and $g(r, \theta) = f(r \cos \theta , r \sin \theta)$. I need help showing that: $$ \left( \frac{ \partial f}{\partial x} \right)^2 + \left( \frac{ \partial ...
0
votes
0answers
9 views

“linear independence” of characters? Artin's theorem.

Let $G$ be a monoid and $K$ be a field. We define a character of $G$ in $K$ as a monoid homomorphism $f\colon G\to K^{\times}$, where $K^{\times}$ denotes the multiplicative group of $K$. One fact ...
-1
votes
0answers
27 views

Let $\operatorname{rank} A=1$ then there are, $x,y\in \mathbb{C}^n$ such that $A=xy^T$ [on hold]

Let $A\in M_n$ and $\operatorname{rank} A=1$. Are there $x,y\in \mathbb{C}^n$ such that $A=xy^T$?
4
votes
3answers
100 views

How do I get $\|x\|\le C\|y\|$ in this case?

I feel that the title is a bit uninformative, please feel free to edit it. This is a problem related to the Open Mapping Theorem. Let $T:X\to Y$ be a bounded linear operator from a Banach space X to ...
5
votes
0answers
22 views

Find the cardinality of a subset of $GL_n( \bf F_p)$

Let $m,n \in \bf N$.Let $\bf F_p$ denote the prime field of characteristic $p$.Consider the set $$ X_m = \{A \in GL_n( \bf F_p): A^m=1 \}$$ Compute the cardinality of $X_m$. Its clear ...
2
votes
0answers
17 views

Why is Gaussian matrix full rank?

Suppose $A\in R^{n\times n}$ is a matrix with independent standard normal entries. Is there an elementary argument to show that $A$ is nonsingular with probability $1$?
1
vote
0answers
30 views

Polar System with Short Answers, How $U(0, \theta)=\pi$ will be calculated?

I read some notes on Laplace. I ran into a short answer question as follows. We have a Laplace equation in Polar Systems: $\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial u}{\partial ...
0
votes
1answer
20 views

Isomorphism is an equivalence relation on finite dimensional vector spaces over $F$.

Show that isomorphism is an equivalence relation on finite dimensional vector spaces over $F$. A relation $R$ is an equivalence relation if it is: reflexive, i.e. $xRx$ for all $x$ symmetric, ...
0
votes
1answer
15 views

How to shift a square into the direction of a given angle so that the old and new borders overlap

Recently I had a problem where I had to shift a square called r into a specific direction, e.g. $45$°. This is what you have: The center of ...
0
votes
1answer
23 views

Elements fail to form a basis

Consider the vector space $P$2 and the set $$5−1t+4t^2,−4+3t+1t^2,8+5t+kt^2$$ For which $k \in \mathbb{R}$, do these three elements fail to be a basis of $P$2? I thought in order to make the three ...
1
vote
1answer
35 views

Why does the additive inverse not follow

I need to prove that the vector space of $\mathbb{R}^2$ with the following operations: $x + y = (x_1 + 2y_1, 3x_2 - y_2)$ The usual scalar multiplication of $cx = (cx_1, cx_2)$ The answers in my ...
1
vote
2answers
43 views

Expressing linear transforms using linear functionals: is this possible?

Work over a fixed but arbitrary field. Let $Y$ and $X$ denote finite-dimensional vectorspaces, and let $y \in Y^n$ denote a sequence of elements of $Y$, where $n$ is a natural number. It seems ...