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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277 views

A proof about polynomial division

Suppose $g(x)=ax+b$,$a,b\in K$,$K$ is a field, and $f(x)\in K[x]$, prove: $$g(x)|f^2(x)\Leftrightarrow g(x)|f(x)$$ The $\Leftarrow$ part is so trivial. But for the $\Rightarrow$ part I get ...
3
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1answer
31 views

If $ S,T \in L(V_{1},V_{2}) $, then show that $ \ker(S) \subseteq \ker(T) $ if and only if $ T = P S $ for some $ P \in L(V_{2}) $.

Let $ V_{1} $ and $ V_{2} $ be finite-dimensional vector spaces over a field $ \mathbb{K} $, and let $ S,T \in L(V_{1},V_{2}) $. Then show that $ \ker(S) \subseteq \ker(T) $ if and only if $ T = P ...
0
votes
1answer
42 views

Matrix transformation: not how to interpret this question

It says "By computing the eigenvalues and eigenvectors of the matrices, give a geometrical description of the linear transformation associated with the matrices" And then i was given 4 2/2 matatrice ...
3
votes
1answer
42 views

If $A$ is normal matrix and $A$ has distinct eigenvalue and $AB=BA$.why $B$ is normal.

If $A$ is normal matrix and $A$ has distinct eigenvalue and $AB=BA$.why $B$ is normal?($A,B \in {M_n}$)
2
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1answer
61 views

What are the differences between the 3 “versions” of “Finite-Dimensional Vector Spaces” by P.R. Halmos?

There are 3 versions on amazon: the 1st one is published at 1993 Aug, by Springer, noted "1st ed. 1958. Corr. 2nd printing 1993 edition (August 20, 1993)"; the 2nd one is published 2014 April by ...
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0answers
41 views

What is meant by an eigenvalue of 2 matrices?

In looking for a way to compare covariance matrices, I came across a paper that formulates a metric using what appears to be a joint eigenvalue. I'm not familiar with this idea. Thus we propose ...
0
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1answer
29 views

Are approximate least square intersections unique?

I seem to be getting a different approximate intersections for the same three lines by multiplying one of the line equations (so that the equation still defines the same line but has different numbers ...
0
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0answers
111 views

Understanding Eigenvector

We have a matrix $A$ of size $N \times M$, where $N\le M$. Consider a vector $V$ of length $N$. Now I take product of $AV$ to get a vector $W$ of length $M$. Here I have projected the original ...
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0answers
45 views

How many distinct possible forms for its Jordan canonical matrix are there? 4x4 non-diagonalizable matrix with two unique eigenvalues

I know the sum of $A_m$ equals $4$ as $\dim(A) = 4$ and sum of $G_m$ can't equal $4$ as $A$ is non-diagonalizable. After I write down all the cases, what should I do?
0
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1answer
197 views

Find the linear transform matrix with respect to an ordered basis.

Let u1=[3,1]^T, u2=[5,2]^T, v1=[1,-2]^T, v2=[1,-1]^T, and let L be a linear operator on R^2 whose matrix representation with respect to the ordered basis u1, u2 is A=[2 1;3 2] ([2,3]^T column 1 and ...
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1answer
26 views

Can't create equations to solve linear algebra problem

Problem Suppose that vectors $x_1, x_2, \dots, x_n$ have the following property: for each $i$ the sum of all vectors except $x_i$ is parallel to $x_i$. If at least two of the vectors $x_1, x_2, ...
0
votes
1answer
42 views

$A \in {M_n}$ is normal.why the null space of $A$ is orthogonal to the range of $A$?.

If $A \in {M_n}$ is normal.why the null space of $A$ is orthogonal to the range of $A$?
3
votes
1answer
37 views

$A \in {M_n}$ is normal.why the range of $A$ and ${A^*}$ are the same?.

Let $A\in {M_n}$ be normal. Why the range of $A$ and ${A^*}$ are the same?
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2answers
42 views

Is the given matrix unitary?

I am given a matrix having blocks of unitary matrices along the main diagonal eg. $$M = \begin{pmatrix}A & 0\\ 0 & D\end{pmatrix}$$ Here $A$ and $D$ are $3$x$3$ and ...
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0answers
409 views

Really confused about LU decomposition and Doolittle algorithm

I'm really confused about the Doolittle algorithm, so I need some help. At the end of the description at wikipedia, it says It is clear that in order for this algorithm to work, one needs to ...
2
votes
1answer
98 views

How to compute coefficients of the Vandermonde polynomial?

I am trying to find the coefficients of the monomials in the expansion of $$\prod_{1\le i < j \le n}^n (x_j - x_i)$$ also known as the Vandermonde determinant. For example, for $n=3$ we have ...
1
vote
1answer
41 views

Given $\sum_{i=1}^{n} \alpha_i f_i=0_E$ prove that $\alpha_1 = 0$

We have $E$ a vector space of functions $\mathbb{R} \rightarrow \mathbb{R}$ Let $a_1 > a_2 > ... > a_n$ be such that $n \geq 1$ . Let $f_1,..,f_n$ be vectors of E such that $\forall x \ \in ...
0
votes
1answer
26 views

Saying that a set is a subset of random elements from another set?

For context, I'm looking to prove one of DeMorgan's Laws (I just started reading AFCLA; I'm on the section where set notation is being introduced). The one I'm trying to prove, in particular, is $$ ...
1
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1answer
44 views

Is it true that linear relations between outer products remains unchanged under change of basis that are full column rank?

Consider a set of outer products $\{ u_i v^T_i \}_i$. I want to show that they are linearly independent. i.e.: $$ \sum^k_{i=1} \alpha_i u_i v^T_i = 0 \iff \alpha_i = 0$$ However, I only know for ...
4
votes
1answer
194 views

Why is $\mathbb{R}^2$ not a subspace of $\mathbb{R}^3$?

I cannot understand why $\mathbb{R}^2$ is not a subspace of $\mathbb{R}^3$. My reasoning is as follows: Choose any elements $v_1$ and $v_2$ from $\mathbb{R}^2$, add them together you get an ...
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0answers
31 views

If A1 and A2 are irreducible row stochastic matrices, can we prove that (I-A1)(A2-I) is stable?

$A_1\in \mathcal{R}^{n\times n}$ and $A_2\in \mathcal{R}^{n\times n}$ are row stochastic (row-sum-1) but not necessarily symmetric, and $I\in \mathcal{R}^{n\times n}$ is the identity matrix. Can we ...
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0answers
35 views

Task about Hartley information(logarithm from cardinality of the set) of a number of paths in a graph and about limit linked with this information.

Let $L_n$ be the number of all paths of length n in a directed graph(below). It is needed to find $lim_{n \to \infty}\chi(L_n)/n$ where $\chi(L_n)$ is Hartley information in $L_n$ set. (If I am not ...
3
votes
0answers
43 views

Convergence of Arnoldi method

I would like to compute the largest real eigenvalue of a matrix in the following form: $$\begin{bmatrix} 0 & I_n \\ P & Q \end{bmatrix},$$ where $I_n$ is the $n \times n$ identity matrix, $P$, ...
0
votes
1answer
31 views

$Span(A)\cap Span(B\setminus(A\cap B))=\{\vec 0\}\Longrightarrow Span(A)\cap Span(B)=Span(A\cap B)$?

$A$ and $B$ are two linearly independent sets. $A\cap B = \varnothing$ and $A\nsubseteq B$,$B\nsubseteq A$. Is the following statement true?: $$Span(A)\cap Span(B\setminus(A\cap B))=\{\vec ...
0
votes
3answers
50 views

Polynomial Question Help

I have the equation $2x^{3}+3x+3=0$ and have been told to substitute in $x=u+2$ which gives $2u^{3}+12u^{2}+27u+25$. I am the told to work out the value of ${1\over \alpha+1} + {1\over \beta+1} + ...
0
votes
1answer
23 views

How do I verify the orthogonal decomposition theorem for a specific matrix?

How do I verify the orthogonal decomposition theorem for $$A = \pmatrix{2 & 1 & 1 & \\ -1 & -1 & 0 \\ -2 & -1 & -1}$$ In other words, I want to show that for all $A \in ...
3
votes
2answers
53 views

How to use Cayley-Hamiltonian theorem in proving upper bound on linear space $W$?

If $W = span(I,A,A^1,A^2, \dots)$. What is the upper bound on dimension of $W$? All matrices are $n \times n$. I know that the dim($W$) $\leq n$, by the Cayley-Hamiltonian theorem. However, I don't ...
4
votes
2answers
76 views

If $BA$ has $-1$ as an eigenvalue, then so does $AB$?

I was just encountered with a rather tough problem as follows: Suppose $A,B\in M_n(\mathbb R)$, prove: $$\det(I_n+AB)\ne0\Rightarrow\det(I_n+BA)\ne0$$ Although at this moment I am still at a ...
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vote
2answers
52 views

Proving rank of matrix product

$A$ and $B$ are $n \times n$ matrix. rank($A$) = $n-1$. rank($B$) = $n$. I want to show $rank(AB) = n-1$. A solution provided to me is: We want to find the N($AB$), the null space of $AB$. To find ...
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votes
4answers
149 views

Square in the complex plane given three vertices. Find the fourth complex number vertice.

There is a square in the complex plane. Four complex numbers form the four vertices of this square. Three of the complex numbers are $-19 + 32i,$ $-5 + 12i,$ and $-22 + 15i$. Find the fourth complex ...
1
vote
1answer
19 views

Matrix of linear application

I'd like to give the matrix in the canonical basis of : $E=\mathbb{K_2}[X]$ and $L:E \rightarrow E$ the linear transformation $p \rightarrow q$ : $\forall x \ \in \ \mathbb{R}, q(x)=xp'(x)+p(x+1)$ I ...
3
votes
4answers
384 views

Why does a singular matrix imply that it does not have a solution?

(Trying to learn linear algebra over here) The augmented matrix in question: $$\begin{bmatrix}0 & 1 &5 & -4\\1 & 4 & 3 & 2\\2 & 7 & 1 & -2\end{bmatrix}$$ So I ...
3
votes
1answer
53 views

Are points in general position generic points?

In Harris' algebraic geometry book, $p_{1},\ldots,p_{r}\in\mathbb{P}^{n}$ are said to be in general position if no $n+1$ or fewer of them are dependent. I want to prove that, if ...
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0answers
76 views

Question on the uniqueness of LU decomposition

Let $A \in \mathbb{K}^{n \times n}$ a matrix, and suppose that we can run the Gaussian elimination on $A$ without row or column interchange, so there exist the $LU$ decomposition of $A$. We define ...
1
vote
1answer
18 views

Prove the size of a linear transformation matrix

I am trying to prove that the size of a linear transformation matrix going from $R^k$ to $R^{p}$ is a $p*k$ matrix. Assuming $ p,k \geq 1$. I can prove it for fixed values of $p$ and $k$ but I am ...
15
votes
1answer
235 views

Linear Algebra : Invertible Matrix Proof

I was doing some linear algebra exercises and came across the following tough problem : Let $M_{n\times n}(\mathbf{R})$ denote the set of all the matrices whose entries are real numbers. Suppose ...
3
votes
3answers
77 views

$T,S: V\to V$, prove that $TS$ and $ST$ have the same eigenvalues

hey I was trying to prove this proposition by dividing to cases and this is what I've got so far: let's assume without loss of generality that T is invertible: ST = [T][S] TS = [S][T] ...
1
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1answer
20 views

Proof of linear transformation $\mathbb{\tilde{L}}(u)=\mathbb{L}(u)$

I'm working on linear transformation and trying to answer : Let $E$ and $F$ be two vector spaces on $\mathbb{K}$, $E$ is finite. $V \ \subset E$ a subvector space of E. $L \ \in \mathbb{L}(V,F)$ ...
4
votes
3answers
190 views

Show that the additive inverse condition can be replaced by $0v = v$ for all $v \in V$

In the definition of a vector space, the additive inverse condition requires that for every $v \in V$ (where $V$ is a vector space over $\mathbf{F} = \mathbb{R}$ or $\mathbb{C}$), there exists ...
2
votes
2answers
143 views

When can the rank of a submodule be bigger than the rank of the module itself?

It is well known that the dimension of a subspace is less than or equal to the dimension of the vector space it is contained in. The same is true e.g. for modules over a principal ring. I am looking ...
3
votes
0answers
38 views

Which elements of $su(n)$ commute with those of a subalgebra $su(2)$

Given a subalgebra $su(2) \subset su(n)$ , how many generators of $su(n)$ commute with any element in the subalgebra $su(2)$? I know that there are at least $n-2$ elements in $su(n)$ satisfying this ...
2
votes
1answer
64 views

how to prove if [T]b is diagonal then there is a scalar “a” such that T(v)=av

hey i was trying to prove the next proposition: given T:V->V for every Basis B, if the matrix [T]B is diagonal, then there is a scalar "a" for every v in V such that T(v)=av this is what i managed ...
2
votes
1answer
46 views

Norm of integral operator in $L^1(0,2)$

How exactly do I show that an integral operator is bounded. For example, consider the following operator $$ T: L^1(0,2) \to L^1(0,2)\\ (Tf)(x):=\int_0^x tf(t) dt$$ My first approach \begin{align} ...
3
votes
1answer
68 views

To Show$ Ax=y$ has no Solutions given that A is a 3x3 non invertible matrix

I am trying to answer the following question: Given that A is a 3x3 matrix where the last row is the sum of the first two rows show that$ Ax=y$ has no solutions. $y \in R^3$ I was thinking that ...
2
votes
3answers
63 views

How does this reduced matrix indicate that the vectors are linearly independent?

I know that a set of vectors is linearly independent when a linear combination of them equal to zero is only satisfied by coefficients that are all zero. For this particular question, we have a ...
4
votes
1answer
58 views

Name of the LU decomposition algorithm

On the wikipedia page of LU decomposition there is an algorithm that produce the decomposition. It is called Doolittle algorithm. I'm really interested who is Doolittle? Or from where the name comes ...
2
votes
1answer
35 views

finding eigenvalues of a linear transformation and determine if its diagonalizable

Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that $T(a_1,a_2,a_3) = (-3a_3, a_1+5a_3,a_2-a_3)$. (i) Find all the eigenvalues of T (ii) For each eigenvalue ...
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1answer
39 views

Show positive definite

On the complex vector space $C([0,1], \mathbb{C})$ one defines a scalar product $<f,g>:=\int_0^1 f(t)\bar{g(t)}dt$. How do I show that this bilinear form is positive definite? I. e. how do I ...
1
vote
1answer
59 views

When can vectors of one basis be expressed as linear combination of vectors of another basis with unitary matrix coefficients?

If I have two normalized basis $\{v\}$ and $\{w\}$ for the same hilbert space of dimension $n$ ( not necessarily orthogonal ), then when can we write the following $$v_i=\sum c_{ij}w_j.....(1)$$ such ...
1
vote
1answer
41 views

Mapping of the eigenvector of eigenvalue 1 to a different matrix

Let $M \in (0,1)^{n\times n}$ be an irreducible and primitive column stochastic matrix. Then for the Perron theorem, $\exists x^* : Mx^* = x^*$. We want to build a matrix $K \in \mathbb{R}^{n\times ...