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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1answer
32 views

A linearly dependent subset

Once again, I have a linear algebra question, this one goes as follows: if $A$ is a linearly dependent subset of $\Bbb R^n$ then the dimension of the subspace spanned by $A$ is strictly less than ...
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1answer
240 views

Damped harmonic oscillator Linear Algebra

I'm trying to solve the differential equation for the damped harmonic oscillator doing an ordinary Linear Algebra approach, eigenvalues/vectors, Jordan form and such. I know it's probably overkill to ...
3
votes
0answers
31 views

Jordan basis of $\mathcal{M}_{\mathcal{T}}(A)$

Let $A\in M_{n\times n}(\mathbb{R})$ be a matrix. Let $\mathcal{B}$ be a basis of $\mathbb{R}^n$ and $X:=\mathcal{M}_{\mathcal{B}}(A)$. If $\mathcal{S}$ is the basis for which ...
0
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1answer
28 views

Specify any perspective transformation

I want to specify a perspective transformation using rotation angles. This should transform a homogeneous co-ordinate $q$ to another. Basically, a square would look like a trapezoid after the ...
0
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1answer
44 views

Finding eigenvalues of a matrix

Given matrix $A = \begin{bmatrix} 0 & 0 &2 \\ 0 & 2 & 0\\ 2 &0 &0 \end{bmatrix}$ I found its eigenvalues: $\lambda_{1}=2, \lambda_{2}=2, \lambda_{3}=-2$. With eigenvalue ...
2
votes
1answer
893 views

How to solve a 3x4 matrix has no solution, a unique solution, and infinite solutions??

The system is : $$ \begin{matrix} 1 & -4 & 6 & a & | & 0 \\ -2 & 5 & -4 & -1 & | & b \\ 1 & -10 & 22 & 8 & | & c \end{matrix} $$ After ...
0
votes
1answer
60 views

what is matrix : each row vector is rotated two(and more such $k$) element to the right relative to the preceding row vector?

In linear algebra, if each row vector is rotated one element to the right relative to the preceding row vector,we call this matrix is a circulant matrix (is a special kind of Toeplitz matrix) ...
2
votes
2answers
74 views

Let $(V,\vert\vert\cdot\vert\vert)$ be a Banach space. Prove if $W\subset V$ and $\dim(W)=n$ then $W$ is a closed subset of $V.$

The original (with words) problem statement is: Let $(V,\vert\vert\cdot\vert\vert)$ be a Banach space. If $W$ is a finite dimensional subspace of $V$ then $W$ is a closed subset of ...
0
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1answer
120 views

Condition number vs. reconstruction error

Suppose I want to solve a simple, linear inverse problem given by $\mathbf{y} = \mathbf{A} \cdot \mathbf{c}$ where $\mathbf{A}$ is an $M \times K$ matrix and I want to solve for $\mathbf{c}$ ($M$ = ...
0
votes
1answer
34 views

Linear transformation: projection

Let $V=\{(x_1,x_2,x_3):x_1-x_2+x_3=0\}$ find the standard matrix for the projection on $V$ Then $S=span[(1,1,0),(-1,0,1)]$ So, I know that I have to evaluate the cannonical vectors $e_1=(1,0,0) ...
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votes
1answer
91 views

Differentiation of the transpose of a vector? [closed]

Suppose $s$ is a scalar, and $x$ is a vector, how would I calculate $$ \left(\frac \delta {\delta x} (x^T s)\right) $$Basically I couldn't find any reliable source letting me know how to ...
0
votes
2answers
165 views

Norm of a matrix and lower bound for its determinant

Assume that $M$ is a positive constant, $A=[a_{ij}]$ is a matrix, and $\vert a_{ij}\vert \geq M $ for all $1\leq i,j \leq n$. Also, assume that $\det(A) \neq 0$ .Can we conclude that there exists a ...
1
vote
0answers
70 views

Monotonically increasing maximum eigenvalue

Let a matrix $A \in \mathbb{R}^{n \times n}$ be the convex combination of two matrices as $A = qB + (1-q)C$. Define $B$ as unit anti-diagonal. Define $C_{i,j} = \delta_{i,i+1}$. Consider $A$ for ...
0
votes
1answer
184 views

Find basis of the annihilator set

$V$ $= \text{span}\{(1,2,3),(1,1,1)\}$ $\subseteq \mathbb{R}^3$. Find the vectors spanning $V^0$ in terms of the usual basis for $(\mathbb{R}^3)^*$. So we want linear functionals $f \in V^*$ such ...
2
votes
2answers
718 views

Solve the following differential equation: $xy' - y = x^2$

I'm preparing to exam in Linear Algebra $2$ and I have problems with differential equations.. For example, the following exercise: Solve the following differential equation: $xy' - y = x^2$. I ...
0
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1answer
38 views

What do these terms actually mean about linear transformation?

In a book it is written " Let A be a fixed $m\times n $ matrix with entries in the field $\mathbb F$. The function $T$ defined by $ T(X)=AX$ is a linear transformation from $\mathbb F^{n\times 1 }$ ...
0
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1answer
54 views

Eigenvectors of (0,0,0)

Find the eigenvectors of $$\mathbf{A}=\begin{bmatrix} 2 & -1 & 2 \\ 5 & -3 & 3 \\ -1 & 0 & -2 \end{bmatrix}$$ Eigenvalue are $-1$ (because of repeated roots). I find ...
0
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0answers
48 views

Understanding 2nd half rank-nullity theorem proof.

I'm trying to understand the second half of the rank-nullity theorem (the part that shows $T(e_{k+1}) \dots T(e_{k+r})$ is independent). Assume $e_1 ,\dots e_k, e_{k+1}, \dots e_{k+r}$,is a basis for ...
6
votes
1answer
184 views

Artin exercise on free modules homomorphism

2.3 Let $A$ be the matrix of a homomorphism $\varphi:\mathbb Z^n\to\mathbb Z^m$ of free $\mathbb Z$-modules. (a) Prove that $\varphi$ is injective if and only if the rank of $A$, as a real ...
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0answers
59 views

Linear Algebra Matrix Transformation Question

Can someone please help me out with this question. If a nonzero matrix $A$ is transformed from $\mathbb{R}^3$ to $\mathbb{R}^2$, then the null space of $A$ must be a one dimensional (sub)space of ...
1
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1answer
130 views

Inequality for norm of linear combination of linearly independent vectors

I'm trying to find a proof for the following: Let {$u_{1},...,u_{n}$} be a linearly independent set of a normed space $X$. Then, there is a constant $c>0$ such that for every set of scalars ...
0
votes
1answer
110 views

Systems of linear differential equations - eigenvectors

Solve the following system of equations $ \begin{cases} x_1^{'}(t)=x_1(t)+3x_2(t) \\ x_2^{'}(t)=3x_1(t)-2x_2(t)-x_3(t) \\ x_3^{'}=-x_2(t)+x_3(t)\end{cases} $. First, I create the column vectors ...
1
vote
1answer
178 views

Computing Jordan Canonical Form

I know how to compute Jordan Canonical Form of a nilpotent matrix, but I find a little bit tedious and long my method for computing the JCF of a general triangulable matrix. I'll show you how I ...
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0answers
38 views

find error for $AX = B$ for many right hand sides

In order to find error for $AX = B$ for many right hand sides,is the below the right way? Let's say the product $$ AX = \begin{pmatrix} 1 &2\\ 3 &4\\ 5 &6 ...
1
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0answers
56 views

Computationnal geometry: vector, basis, point and coordinate system?

I am trying to build a small geometrical library in C++, that is mathematically consistent (not so false). The goal here is to construct two concepts: vectors and points. I am not sure that the ...
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votes
5answers
71 views

Vector spaces and kernels

Let $V$ and $W$ be vector spaces over $\Bbb{F}$ and $T:V \to W$ a linear map. If $U \subset V$ is a subspaec we can consider the map $T$ for elements of $U$ and call this the restriction of $T$ to ...
0
votes
2answers
55 views

How to expand $(x+ty)^{T}A(x+ty)$

I am trying to expand the expression: $$(x+ty)^{T}A(x+ty),$$ with $x,y$ being vectors and $A$ a matrix. All I know is the distributed law, $(A+B)C = AC+BC$. Can someone explain how to arrive to the ...
1
vote
3answers
348 views

Origin in vector space?

In the wikipedia article about vector space I do not understand this sentence Roughly, affine spaces are vector spaces whose origin is not specified. A vector space does not need an origin. When ...
5
votes
2answers
343 views

Is evaluation homomorphism surjective?

Let $A^n$ be an affine space over $\mathbb{C}$ and let $\mathbb{C}[X_1,\cdots,X_n]$ be the polynomial ring of $n$ variables. Then $A^n\to (\mathbb{C}[X_1,\cdots,X_n])^*$ by evaluation homomorphism, ...
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0answers
57 views

Adjacency matrix of strongly connected digraph

If I have $A$ the adjacency of a strongly connected digraph, I want to show: For $\lambda$ satisfying $Ae= \lambda e$ for nonegative $e$, I want to show for any eigenvector (could be negative), the ...
-1
votes
0answers
39 views

Compute $\hat{x}$ with the given errors $(2,-6,4)$

Q. Suppose the measurements at $t = -1,1,2$ are the errors $2, -6, 4$ in Problem 18. Compute $\hat{x}$ and the closest line to these new measurements. Problem 18: Find the projection $$p = A ...
0
votes
1answer
38 views

how can get simple function

how can get $y$ simple function of $x$ from : $$ \frac{2 \sqrt{y} \sqrt{y-b} \log \left(\sqrt{y-b}+\sqrt{y}\right)}{\sqrt{\frac{A y (y-b)}{b}}}+\text{constant} =x$$ where : $$ y=y_0 at x=0$$ and ...
0
votes
3answers
47 views

A question about eigenvectors.

Let $T\in L(V,V)$, and let $\{v_1,v_2,\dots,v_n\}$ be a basis of $V$ consisting of eigenvectors of $T$, belonging to eigenvalues $a_1,a_2,\dots,a_n$ respectively. Then $Tv_i=a_iv_i$. Prove that ...
0
votes
1answer
47 views

Different solutions of homogenous system

I'm solving a homogenous system. One solution: $$\begin{pmatrix}1 & 2 & -2 & 2 & -1\\ 1 & 2 & -1 & 3 & -2\\ 2 & 4 & -7 & 1 & 1 ...
0
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1answer
59 views

If two matrices are similar, the geometric multiplicities of their eigenvalues are the same

Problem Let $A$ and $B$ be similar matrices. Prove that the geometric multiplicities of the eigenvalues of $A$ and $B$ are the same. [Hint: show that, if $B=P^{-1}AP$, then every eigenvector of ...
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votes
0answers
58 views

Non-orthogonal projector

Suppose an $n$-by-$n$ projector projects onto the range of the linearly independent vectors, $U_i$ ($i = 1,2,...,k$) along the range of a certain subspace $W$ (not necessarily orthogonal to the range ...
1
vote
1answer
98 views

Linear Algebra, meaning of 0 determinant in linear transformations

Lets say the area of a figure in $\Bbb R^2$ was $10$. Then after a noninvertible linear transformation from $\Bbb R^2$ to $\Bbb R^2$, is there enough info to determine the new area? Since its ...
0
votes
1answer
71 views

The following subset is a subspace of $\mathbb{R}^2$

I know that $5x+xy=0$ is a subspace of $\mathbb{R}^2$ since it follows the $(u+v)$ and the $c(u)$ theorem. However, is this subset always a subspace or only sometimes a subspace?
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1answer
34 views

Are all matrices of order $n^2$ bijective linear transformations?

My linear algebra book says that all matrices $T\in L(V,V)$ (of order $n^2$ if the dimension of $V$ is $n$) are linear transformations. Also, each such mapping is bijective. Shouldn't only ...
1
vote
2answers
192 views

Linear Algebra need help with proof please over eigenspaces

I know that if x and y are distinct eigenvalues of an nxn matrix A, then the intersection of eigenspaces is the 0 vector. How can I prove this?
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1answer
51 views

T/F: Any of these two statements imply the third: $A$ is Hermitian, $A$ is unitary, $A^{2}=I$

I think this is false because: Assuming that $A$ is Hermitian, then $A^{H}=A$. Now I assume that $A^{2}=I$, so $A^{H} \cdot A^{H} = I$ Which doesn't imply that $A$ is unitary. So the statement is ...
2
votes
1answer
57 views

Given matrix $B =\begin{bmatrix}0&-1\\0&0\end{bmatrix}$, find $e^{Bt}$ from a short infinite series

Given matrix $B =\begin{bmatrix}0&-1\\0&0\end{bmatrix}$, find $e^{Bt}$ from a short infinite series. Also check that the derivative of $e^{Bt}$ is $Be^{Bt}$ I found that the eigenvalues of ...
0
votes
1answer
48 views

How to simplify floor polynomial given lower bound on x?

$$ \left\lfloor\frac{8x^2 + 5x -4}{3x^2 + x}\right\rfloor $$ where $x$ > $\sqrt{8}$ How would you simplify this type of expression? *Please note the floor operation surrounding the expression ...
0
votes
3answers
958 views

Reflection across the plane

Let $T: \Bbb R^3 \rightarrow \Bbb R^3$ be the linear transformation given by reflecting across the plane $S=\{x:-x_1+x_2+x_3=0\}$ (...) Then, $S=\gen[(1,1,0),(1,0,1)]\$ But how can I get the matrix ...
1
vote
2answers
81 views

Linear transformations and change of basis [duplicate]

Let $T:\in \Bbb R^3 \rightarrow \Bbb R^3$ be the linear transformation given by reflecting across the plane $-x_1+x_2+x_3=0$ (...) If $S=\{x:-x_1+x_2+x_3=0\} \implies S=\gen[(1,1,0),(1,0,1)]$ But ...
0
votes
1answer
885 views

Markov matrices: finding the initial state vector

I am wondering how can I find an initial state vector for this problem. If the air quality is good one day, it has 95% chance it will be good the next day. If the air quality is bad one day it has ...
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0answers
83 views

Hermitian and self-adjoint operator.

I was trying to understand why this operator is hermitian (I see that) but not self-adjoint: We have the operator $P:\phi(\cdot )\mapsto -i\phi '(\cdot )$, dfined in $C_0^\infty (I)$ (infinitely ...
0
votes
1answer
47 views

Finding a Householder matrix for row elimination

I was wondering how to find a Householder matrix such that I could apply it from the right side of a matrix and eliminate values along a row. For example, I have a matrix of the form B = ...
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3answers
43 views

Linear Algebra, find determinant with x1, x2,…,xn as scalars

I have no clue how to even begin solving for $\det(A)$ since $n$ is unknown, HELP!
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1answer
39 views

$A \in L(V)$ where $V$ is a FDVS. Prove that there is an invertible $Q$ such that $AQ$ is a projection.

On P. 94 of Halmos' FDVS, you will find the following theorem: Corresponding to any linear transformation $A$ on a finite-dimensional vector space $V$, there is an invertible linear transformation ...