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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T has zero as a characteristic root [on hold]

Let $ V $ be a vector space over field $F$. $T$:$V$$\rightarrow$$V$linear transformation such that $T$ has zero as a characteristic root.Then 1.T is diagonalisable over $F$ 2.Multiplicity of each ...
0
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0answers
38 views

Subset of vector space containing zero vector. [on hold]

Subset of vector space containing zero vector is always linearly independent.Is this statement is true?
0
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2answers
36 views

Proof that Every Positive Operator on V has a Unique Positive Square Root

Suppose V is a finite-dimensional, nonzero, inner-product space over F, and F denotes R or C. My thought is : suppose T is a positive operator; thus, T is self-adjoint. Every self-adjoint operator on ...
4
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0answers
26 views

Reference for Generalized Eigenvectors

I am looking for references on generalized eigenvectors and Jordan matrix representation. I would like a brief but complete introduction of this concepts with a nice treatment of the most important ...
2
votes
2answers
73 views

Prove there is a subspace of $V$ isomorphic to $T(V)$

If $T:V\to V$ is a linear transformation and $T(V)$ is of finite dimension then prove that there is a subspace $U$ of $V$ isomorphic to $T(V)$ and then show that, if $x,y\in V$, then $(x+U)\cap ...
3
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2answers
55 views

Is calling a linear-equation a linear-function, misnomer or completely wrong?

From my college life, I remember many professors used to call a linear-equation a linear-function, however: A standard definition of linear function (or linear map) is: $$f(x+y)=f(x)+f(y),$$ ...
0
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1answer
22 views

Identifying if a $ S $ given is a vector subspace

Could you help me to identify if $ S $ is a vector subspace? I started learning linear algebra and I get this question and I am lost.
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2answers
58 views

Show that vectors of the form (a,b,1) do not form a vector space

Show that vectors of the form $(a,b,1)$ do not form a vector space I tried using the vector space axioms to attack the problem but I am not going anywhere with this problem. I do not need a ...
3
votes
1answer
41 views

Rank of the product of 3 matrices

Suppose I have 3 n by n matrices $A,B,C$ with $ABC=0$, what could be the maximal rank of $CBA$? I guess the answer would be n but I failed to prove it( tried to use Rank-Nuillity Theorem but I don't ...
1
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2answers
48 views

If $\operatorname{rank}(A)=n$ then $\operatorname{rank}(AB)=\operatorname{rank}(B)$

I have looked here, but still I cannot understand how to get to equality. Let assume that the matrices are squared $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ is easy to show, but how can I ...
0
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0answers
35 views

How many Geese were there before any flew away? [on hold]

This equation represents geese flying away in one hour intervals. How many geese were there before any flew away? The first part x - [1/5 = x] represents the quantity of geese flying away at 1:00pm ...
0
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1answer
22 views

Inverse Matrix Multiplication

Let $A \in F^{n*n}$ a inverse matrix and $B\in F^{n*n}$ a none inverse matrix We can say that because A is row equivilate to $I_n$$ \implies $ $AB$ is none inverse matrix?
0
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0answers
37 views

Sum/diff of matrix units

I understand what the product of matrix units means, but I don't understand what the sum/difference of two different matrix units represents. For example, what does ${e_{2,2}}-{e_{5,5}} $ equal? ...
3
votes
1answer
43 views

Special solutions to Ax = 0

I solved most of it, just not sure about one point. The problem statement, all given variables and data Suppose A is the matrix shown below: $$ \begin{pmatrix} 0 & 1 & 2 ...
2
votes
1answer
34 views

Vector spaces - Multiplying by zero vector yields zero vector.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space axioms. Axiom ...
2
votes
1answer
63 views

Proof $p(A)=0$ without Cayley-Hamilton theorem when $A$ is upper triangular

I need help proving $p(A)=0$ without Cayley-Hamilton theorem when $A$ is upper triangular, as part of the proof of the Cayley-Hamilton theorem The result makes a lot of sense but I can't prove it ...
1
vote
1answer
45 views

A subset that is closed under multiplication but not addition? [duplicate]

I can't get my head around subspaces despite having studied on them quite a lot. Here goes: The problem statement, all given variables and data Give an example of a non-empty subset U of R^2 such ...
3
votes
1answer
89 views

Changing local coordinates on a manifold by a diffeomorphism

This is the set up of my problem: Let $M$ be a manifold with local coordinates $x^1,\dots, x^n$. Let $x^1,\dots,x^n,\xi_1,\dots,\xi_n$ denote the induced coordinates on $T^\ast M$. Let $f:M\to M$ be ...
1
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0answers
36 views

Vector spaces - Multiplying by $-1$ yields inverse element of vector addition.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is based on vector space related axioms. Axiom ...
0
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0answers
20 views

Rank Factorization

I read the following proof Let A, B be m × n matrices. Then rank(A + B) ≤ rankA + rankB. Proof: Let A = XY,B = UV be rank factorizations of A, B. Then A + B = XY + UV = [X,U][Y V] Therefore, ...
0
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1answer
25 views

Subspace of a vector space Definition

If $W$ is a subspace of a finite-dimensional vector space $V$, then: $\dim(W) \leq \dim(V)$. That makes me think about the definition of a subspace. For example, in $\Bbb R^3$, is $\Bbb R^3$ ...
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0answers
45 views

Matrices over field with characteristic $p$

For $A,B$ $n\times n $ matrices over a field $F$ with characteristic $p$ if $AB-BA=cI$ for $c\in F$ does this imply that $c=0$? Intuitively I would say that it doesn't but I cannot think of a ...
0
votes
2answers
63 views

Compute an upper bound on generalized eigenvalues (by using the coefficients)

Consider the generalized, symmetric eigenvalueproblem: \begin{equation} A x = \lambda B x, \end{equation} with $A, B$ symmetric and $B$ being positive definite. For some computations, i was trying ...
2
votes
1answer
26 views

Find the kernel of linear transformation

Linear transformation $l:\mathbb{R}^3 \mapsto \mathbb{R}$ is determined as follows: $l(1,0,0)=1$; $l(1,4,0)=-1$; $l(0,0,1)=1$. I need to find $\text{Ker}(l)$. Answer should be ...
0
votes
1answer
39 views

What is the space spanned by $a\cos x + b\sin x$?

Consider the case when $x = \pi/4$. $\cos \pi/4 = 1 = \sin \pi/4$. Now, if $a = 1$ and $b = -1$, $a\cos x + b\sin x = 0 $(for non-zero a and b). Does this imply that $\cos x$ and $\sin x$ ARE ...
1
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0answers
32 views

Circular Matrix Linear Independence

Suppose I have the following $N\times N$ circular matrix: $$ \left[ \begin{matrix} 0 & 1 & 2 & .... & N \\ N & 0 & 1 & .... & N-1 \\ . & . ...
0
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1answer
19 views

Lie algebra for SO(3) as a skew symmetric matrix

How can I show that the associated lie algebra for SO(3) is the set of all 3 dimensional skew-symmetric matrices?
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0answers
17 views

Every polyhedron $P \ne \mathbb{K}^n$ equals an intersection of finitely many half spaces.

Currently, I am reading some lecture notes on linear optimisation. I cannot see why the following (seemingly trivial) proposition holds. (How could I understand/proove it?) Every polyhedron $P \ne ...
0
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2answers
69 views

Visually understanding the formula for the distance from a point to plane.

Ok, so we know that if we have an arbitrary point, $p$, and a normal perpendicular to an arbitrary plane, $n$, the distance from the point to the plane can be computed as follows: $$distance = p ...
3
votes
2answers
47 views

What is the determinant value of $J-I$ if $I$ is identity matrix and $J=(1)_{101\times 101}$? [duplicate]

Let $J$ be a matrix of order $101\times 101$ which each entry is 1 and suppose $I_{101}$ is identity matrix of order $101\times 101$. The question is : what should be the determinant value of $J-I$ ? ...
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1answer
59 views

Clarification on this proof from Linear Algebra Done Right 7.41 [on hold]

=========================== For the Yellow highlighted part. I don't understand how do we get to forth equality(marked 2) from third equality (marked 1) For the Red highlighted part. I don't ...
0
votes
3answers
26 views

Prove that operator on $\Bbb R^2$ of counterclockwise rotation is isometry

An operator on $\Bbb R^2$ of counterclockwise rotation (centered at the origin) by some angle is an isometry.
0
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1answer
24 views

Four Fundamental Spaces Question

Why do the four fundamental sub spaces come from Ax=b and A'y=f? Why these two equations?
1
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1answer
27 views

Solving a linear equation with no y

I have run into a problem with a linear equation in which y ends up being cancelled out leaving only 0. In this problem, I am supposed to find the slope and the y intercept. I understand how to find ...
0
votes
2answers
40 views

Curve length under Linear Transformation

On this mathoverflow post, we can see that from $T : \mathbb{R}^n \to \mathbb{R}^n$ linear tranformation with matrix $A$ and a measurable set $S \subseteq \mathbb{R}^n$, we have: $$ \mu(T(S)) = ...
0
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1answer
27 views

Idea for a proof that a Hamel basis is infinite without using a countability argument

I am trying to prove that a Hamel basis is infinite without using the countability argument. My idea goes like this: Assume the basis is finite with irrational elements $a_1,a_2,a_3, \dots, a_n$. I ...
3
votes
1answer
29 views

How to interpret tensor form PDE in terms of matrix algebra

From this mathwork page "c for system", the usual second order PDE is written in tensor form: $$ -\nabla\cdot(\mathbf{c} \otimes \nabla \mathbf{u})+\mathbf{a}\mathbf{u}=\mathbf{f} $$ and ...
1
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0answers
25 views

Let $T$ be multiplication by the matrix $A$ and find basis, kernel for the range of $T$

Let $T$ be multiplication by the matrix $A$. $$A =\begin{bmatrix} 1 & 0 & 3\\ 1 & 2 & 4\\ 1 & 8 & 25 \end{bmatrix} $$ My task is to find a basis for the range of $T$, and a ...
2
votes
3answers
36 views

Defining Linear Transformations

I am currently stuck on a problem (this is not a homework problem) mainly because I am weak at DEFINING functions. The problem states: Suppose $U$ is a subspace of $\Bbb R^3$ defined as $$U=\{(x,y,z) ...
2
votes
3answers
45 views

Determinant-like expression for non-square matrices

I'm interested in whether for any real matrix of size $m \times n$ there is a real number with the following properties: It is a polynomial expression with real coefficients in the entries of the ...
0
votes
1answer
50 views

Enlarge 3 Circles about the same factor to find the Intersect

I currently have 3 circles that not intersect at all. Like this: Now i would like enlarge the circles about the same factor to find the intersection of this three circles. I have tried following ...
0
votes
2answers
39 views

Substitution vs Elimination in Solving Systems of Equations

When solving systems of equations, is it more efficient in terms of time to solve it using substitution or elimination, and what are your reasons for saying so?
0
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1answer
23 views

Solve this equation 3x3 using Gauss-Jordan

I have this problem: With this system, determine the values of $K$ to the system have: a) One only solution b) Don`t have solution c) Infinite solutions $x-3z=-3$ $2x+Ky-z=-2$ $x+2y+Kz=1$ How do ...
0
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0answers
35 views

Solve linear equations [on hold]

\begin{bmatrix} 0 & 0& 1& 1& 1&0 \\ 0 & 0& 0& 0& 2&1 \\ -3 & 0 & 0 & 0 & -2& 0\\ 4& 4& 0& 0& 0 & -1\\ ...
2
votes
1answer
72 views

Why we use $\mathbb{R}^{m \times n}$ notation instead of $\mathbb{R}^{n \times m}$?

I just realised, that I use all the time the notation $\mathbb{R}^{n \times m}$, and all books and papers use $\mathbb{R}^{m \times n}$. $\mathbb{R}^{n \times m}$ is more sympathetic for me, because I ...
1
vote
1answer
47 views

Prove $T$ is a linear transformation and find $\ker T$ and $\mathrm{im} T$

Let $P_n$ be the vector space of all real polynomials of degree less than or equal to n. Let $T : P_3 \rightarrow P_3 $ be defined by $T(p(x)) = xp'(x) - 3p'(x)$, where $p'(x)$ represents the formal ...
3
votes
3answers
105 views

Eigen values of AB and BA

let A be a linear transformation from $R^n$ to $R^m$, and B be a linear transformation from $R^m$ to $R^n$, it's easy to show that AB and BA has same eigen-value(except $0$). But my question is how ...
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votes
3answers
52 views

Is the basis of null space of a matrix always a subset of the basis of its column space?

Given an $m\times n$ matrix $A$, is the basis of its null space (set of $x$ such that $Ax=0$) always a subset of the basis of the row space of $A$? In general, the basis of a subspace may not be a ...
8
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0answers
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+50

Families of Idempotent $3\times 3$ Matrices

I did the following analysis for $2\times2$ real idempotent (i.e. $A^2=A$) matrices: $$ ...
2
votes
4answers
72 views

Showing $U\otimes (V\oplus W) \cong U\otimes V \oplus U\otimes W $

I have to show $U\otimes (V\oplus W) \cong U\otimes V \oplus U\otimes W $ where $U,V,W$ are $K-$vector spaces. One way to give a linear map from left to right is: $$u\otimes (v,w)\mapsto (u\otimes v, ...