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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4
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1answer
22 views

Find the matrix relative to the standard bases

Let $T:P_2 \to P_3$ be the linear transformation defined by $T(p(x)) = xp(x).$ Find the matrix for $T$ relative to the standard bases $B = \{u_1, u_2, u_3\}$, $B' = \{v_1, v_2, v_3, v_4\}$. $u_1 = ...
2
votes
1answer
43 views

Linear Algebra Basis and Dimension

Find a basis for the given subspace by deleting linearly dependent vectors. $S = \text{span}\{(0, 0, 0), (9, 0, 0), (8, 1, 0), (1, 8, 9)\}$ I do not understand how to "delete linearly independent ...
3
votes
1answer
32 views

Characterize magic matrices in terms of their eigenvalues. A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$.

A Magic Matrix over a field $F$ is a square matrix whose row and colums sums $c\in F$. Characterize magic matrices in terms of their eigenvalues. I know that $c$ is an egenvalue and $[1,...,1]^T$ is ...
-1
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1answer
36 views

About summer course or online course of Linear algebra and real anyasis

I just looking for the online course for Linear algebra or real analysis but it should be upper level. i saw MIT and another college but our university said it was not upper level its likely ...
1
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1answer
23 views

In $\mathscr{V}$, let $X \subset \mathscr{V}$ be a set of $n$ vectors. $Y \subset X$ contains vectors all scalar multiples, $X$ linearly dependent.

I would just like to verify that my proofs are sound and receive any suggestions on rewording. (If relevant, I am self-studying and haven't done a serious proof in about a year.) $\mathscr{V}$ is a ...
1
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1answer
9 views

Let $D$ be a nonsingular diagonal matrix. Show that $1\notin spec(DA)$ if and only if $D - A$ is nonsingular.

Let $F = \mathbb{F}_3$ and let $n$ be a positive integer. Let $D = [d_{ij} ]\in\mathscr{M}_{n×n}(F)$ be a nonsingular diagonal matrix and let $A\in\mathscr{M}_{n×n}(F)$. Show that ...
0
votes
1answer
32 views

Questions about Eigenspace

I'm learning about Eigenspaces and have a few questions. Do eigenspaces, eigenvalues, and eigenvectors correspond to a tranformation or can a single vector space $V$ have an eigen-stuff? Is an ...
0
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2answers
15 views

Reflect on y axis in 3D Matrix?

I have a question saying "Define a 3D Matrix that performs a reflection in the y axis" but I don't know how to solve it. So if we have a 2D matrix and we say 'reflection on the y axis' we mean that x ...
1
vote
3answers
41 views

Understand the definition of a vector subspace

I'm pretty new to Linear Algebra and I have started on Vector Spaces. I understand that a Vector space V over the set of real numbers is a set equipped with two operations, namely vector addition and ...
0
votes
3answers
10 views

Give two matrices whose column spaces contain the column space of the given matrix.

Let $$B = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\text{.}$$ Give two matrices whose column spaces contain $C(B)$, the column ...
0
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0answers
10 views

Relation between Ill-posed problem and eigenvectors?

This question is related to the question below: Is there a relation between Ill-posed problems and Eigenvectors. In the answer of the above question, it was shown that the ill-posed problem can be ...
2
votes
0answers
26 views

Dependence of linear algebra theorems of the commutativity of the field.

In the linear algebra course I took vector spaces where introduced with a (commutative) field. The classical theorems are proven under this assumption. However, I was wondering what implications it ...
2
votes
2answers
21 views

Orthogonal projection and subspaces

Consider the vector space $\mathbb{R}^m$ with usual inner product. Let $S_1$ and $S_2$ subspaces of $\mathbb{R}^m$ , $P_1\in\mathbb{M}_m(\mathbb{R})$ a orthogonal projection matrix on subspace $S_1$ ...
0
votes
1answer
11 views

Is there any relation ill-posed problem and not Normal matrix?

I am trying to understand different aspect associated with ill-posed problem. Can we claim that an ill-posed problem $Ax=b$ means that the matrix $A$ is not normal? Further, can we claim that if $A$ ...
1
vote
0answers
24 views

Orthogonal projection matrix proof

Let $P\in \mathbb{M}_m(\mathbb{R})$ a orthogonal projection matrix. Show that the matrix $Q=I-P$ is a orthogonal projection matrix. Make a geometric interpretation of the elements $z=Pb$ and ...
1
vote
0answers
9 views

Showing the action of $SO(p,q)$ in the punctured cone of isotropic vectors is transitive

Consider a real vector space $T$ of dimension $p+q$ with a non-degenerate symmetric bilinear form, $B:T\times T\to\mathbb{R}$, with signature $(p,q)$. Define the cone $$ ...
1
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3answers
35 views

Show that a linear transformation $T$ is one-to-one

Problem: Consider the transformation $T : P_1 -> \Bbb R^2$, where $T(p(x)) = (p(0), p(1))$ for every polynomial $p(x) $ in $P_1$. Where $P_1$ is the vector space of all polynomials with degree less ...
0
votes
0answers
40 views

A Theorem on Skew-Symmetric Bilinear Forms in Hoffman and Kunze's Book

On pg. 377 in Hoffman and Kunze's Linear Algebra(Second Edition) Theorem 6 reads: Let $V$ be an $n$-dimensional vector space over a subfield of the field of complex numbers, and $f$ be a skew ...
1
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2answers
26 views

Finding a Basis for this subspace

Set $V=\mathbb{R}^{2x3}$ and let $U$ be a subspace of $V$ defined by: \begin{equation*} U=\{B=(b_{ij})\in V\mid b_{11} + b_{12} + b_{13} = -4(b_{21} + b_{22} + b_{23})\}. \end{equation*} I would ...
3
votes
1answer
64 views

Origin of the term dual space?

Basically, why is a dual vector space called as such? Is the reason for the term "dual" simply because the two vector spaces are related by a one-to-one mapping, or is there something more to it? ...
1
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1answer
10 views

Expressing problems in canonical form for solving with simplex

The Picnic Hamper Company has a store containing 10,000kg of nuts, 4000 packs of smoked salmon, 2000 bottles of wine and 1500 Victoria sponges. It intends to use these goods to make up three different ...
0
votes
2answers
45 views

How do you find the value of n in this example

$$n^{n-2} = 16$$ I know $n = 4$ through trial and error but how do you find $n$ in a conventional manner? I'm basically trying to solve how many nodes are in a tree that has $16$ spanning trees ...
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votes
1answer
30 views

Do all n x n matrices over the reals represent linear transformations?

Do all $v \in M_n (\mathbb{R})$ represent linear transformations? To add to that a bit to further clarify for myself: Looking up the def. of a transformation it is any function $f$ mapping a set $X$ ...
0
votes
1answer
21 views

Prove the surjectivity of this injective linear map

I am working on the following problem. Let $g : V\to V$ be linear and injective, where $V$ is a vector space over the field K. Prove that, if $V$ is finite-dimensional, then $g$ is surjective. In an ...
3
votes
0answers
40 views

Let $\mathbb{K} $ be a field of characteristic $p>0$ and $\mathbb{F} | \mathbb{K} $ a finite and separable extension.

Let $\mathbb{K}$ be a field of characteristic $p>0$ and $\mathbb{F}/ \mathbb{K}$ a finite and separable extension. Show that if $B=\{\alpha_1,\dots,\alpha_n\}$ is a basis, then ...
0
votes
1answer
15 views

Injective linear endomorphism of hilbert space is bijective?

Is it true that an injective continuous endomorphism of a hilbert space is bijective? If not, are there conditions that imply this? I know this would follow from the rank nullity theorem in finite ...
1
vote
1answer
6 views

Basis for row space of matrix: REF vs. RREF.

When finding a basis for the row space of a matrix, I reduce the matrix to row echelon form, and find the rows that have pivots in them. Does it matter wether you use the echelon or the reduced ...
2
votes
2answers
48 views

$x^n + y^n = z^n$, $n>1$ To show that $x,y,z$ is greater than $n$

Problem: If $x$,$y$,$z$ and $n>1$ are natural numbers with $$x^n+y^n = z^n$$ then show that x,y and z are all greater then $n$. My approach, from Fermat's Theorem we know that $x^n + y^n = z^n$ ...
2
votes
1answer
24 views

Proving a matrix $A$ is of certain form

Let $A\in M_n(\mathbb{C})$, and $A=A^3$, prove that $A^2$ is of form $\begin{pmatrix} I_r & 0\\ 0 & 0 \end{pmatrix}$ where $1\leq r\leq n$. It make sense. My initial thought was to say that ...
1
vote
2answers
12 views

sum of matrices with unique solutions

Let $K$ be any field with a characteristic, different than 2, and $A$ any $n \times n$-matrix over $K$. For the equation $A = B + C$, where $A$ and $B$ are $n \times n$-matrices over $K$, are $B = ...
2
votes
3answers
15 views

Completing an orthonormal basis of a plane to a basis for $\mathbb{R}^3$

I was asked to find an orthonormal basis for the plane $x + 2y +3z =0$. I found a regular basis, $(-2,1,0),(-3,0,1)$, and then performed the Gram-Schmidt process to find 2 orthonormal vectors that ...
2
votes
1answer
26 views

Why is the the double dual functor on finite-dimensional vector spaces naturally isomorphic to the identity?

$\require{AMScd}$ Note: I have already seen this question, which asks about a specific aspect of the construction - here I am trying to construct this functor and failing at a very different stage. ...
1
vote
2answers
55 views

A Linear Operator of Rank 1

Let $T$ be a linear operator with rank $1$ on a finite dimensional vector space $V$.Then Which of the following are true? 1)either $T$ is diagonalizable or $T$ is nilpotent. 2)$T$ is both ...
0
votes
3answers
31 views

Find the complex eigenvectors, knowing the eigenvalues

If $$A= \begin{pmatrix}1 & -1 \\ h^2 & 1\end{pmatrix},$$ I know the complex eigenvalues are $1+ih$ and $1-ih$. How do we find the complex eigenvectors? Can someone please explicitly show me ...
0
votes
1answer
30 views

Properties of a matrix that shares the set of real eigenvalues with its inverse

For a $3\times 3$ real matrix, let $c(A)$ denotes the set of real eigenvalues of $A$. Suppose $c(B)=c(B^{-1})$ for a non-singular matrix $B$ with no repeated eigenvalues. Then which of the following ...
3
votes
1answer
60 views

Show that there exists a vector $v$ such that $Av\neq 0$ but $A^2v=0$

Let $A$ be a $4\times 4$ matrix over $\mathbb C$ such that $rank A=2$ and $A^3=A^2\neq 0$.Suppose that $A$ is not diagonalisable. Then Show that there exists a vector $v$ such that $Av\neq 0$ but ...
0
votes
0answers
13 views

Maximum number of independent parameters for defining a subspace of a vector space

Consider a subspace $W$ in a vector space $V$. The basis of $W$ is a funciton of a set of parameters $\{\alpha_i\}$. What is the maximum number of independent parameters for fully defining the ...
1
vote
1answer
22 views

How to express outer sum in a matrix form?

So I have the following equation for a matrix $\mathbf{B}$ given $\mathbf{A}$: $$ b_{ij} = \sum_k \sum_l a_{ki} a_{jl} $$ The question is if there is anyway that I can write that one compactly in ...
1
vote
1answer
22 views

If a Bilinear Form is Non-Degenerate on a Subspace $W$, then $V=W\oplus W^\perp$.

$\newcommand{\range}{\text{image}}\newcommand{\ann}{\text{Ann}}\newcommand{\set}[1]{\{#1\}}$ Problem: Let $V$ be a finite dimensional vector space over a field $F$ and $f$ be a symmetric bilinear ...
0
votes
1answer
26 views

Find the vectors $x$ such that $T(x) =x$ [on hold]

I'm provided with a matrix $T$ which is $[2 -3; -1 4]$ and as the title says I'm supposed to find a vector $x$ such that when I multiply $T$ by it, $x$ is the result. The problem seems simple enough ...
1
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2answers
30 views

linear algebra-norm of matrix

Why $ \|A\| = \|A^*\| $ in matrix ? Suppose that A is a normal matrix. I know $ A^* = A^{-1} \det(A) $ and so $\|A^*\| = \|\det(A) A^{-1} \| \rightarrow \|A^*\|=\det(A) \|A^{-1}\|$ but I can't prove ...
1
vote
2answers
36 views

Vectors in an inner product space

Let $u,\,v,\,w$ be the vectors in an inner product space $V$, satisfying $\|u\|=\|v\|=\|w\|=2 $ and $\langle u,v\rangle=0,\langle u,w\rangle=1,\langle v,w\rangle=-1$.Then which of the following are ...
1
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0answers
22 views

Generate a random neutrally stable matrix

I need to generate random real matrices such that all eigenvalues have real part equal to 0 -- i.e. random neutrally stable matrices. What's the simplest way to do this? Note that I don't care about ...
1
vote
0answers
20 views

Eigenvalue multiplicity of a product of two real skew-symmetric matrices

All the roots of characteristics polynomial of $AB$, where $A$, $B$ are skew symmetric matrices of order $2n$, are of multiplicity greater then $1$. I know that eigen values of skew symmetric ...
0
votes
2answers
23 views

The angle between $u$ and $v$ is $30º$, and the vector $w$ of norm $4$ is ortogonal to both $u,v$. Calculate $[u,v,w]$.

The angle between the unit vectors $u$ and $v$ is $30º$, and the vector $w$ of norm $4$ is ortogonal to both $u,v$. The basis $(u,v,w)$ is positive, calculate $[u,v,w]$. I did the following: ...
9
votes
3answers
122 views

If $\,A^3-A+I=0,\,$ then $A$ is invertible

Prove or disprove. If $A$ is a square matrix and $A^3-A+I=0,$ then $A$ is invertible. Is it possible to say the characteristic polynomial of $A$ is $t^3-t+1=0$ and $A$ is invertible since $0$ is not ...
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votes
1answer
26 views

Representing a series of Matrix inner product with a single matrix product.

I have a set of constraints in my optimization problem, constraints in the form , $\langle A, e_i e_j^T \rangle = r_{ij} ,\forall i,j \epsilon S$, where $A$ is an $n*n$ semidefinite and symmetric ...
-4
votes
1answer
25 views

A $5 \times 5$ matrix [on hold]

Let $A$ be $5 \times 5$ matrix. The dimension of its column space is $3$. Is it possible that $\det(λ−A) =λ^3(λ−1)(λ−2)$? Please explain what this statement means and the answer to it.
5
votes
2answers
71 views

relation between eigenvectors of $A$ and $A^TA$?

Is there a relation between the eigenvectors of a linear operator (Matrix) $A$ and eigenvectors of $A^TA$? This question is related to eigenvectors ans not eigenvalues. Further the size of the matrix ...
0
votes
1answer
39 views

Linear transformations and their kernels

Am I correct to assume all of the following are linear transformations? I tested all 3 for the 2 conditions $T(A_1+A_2) and T(kA)$ but I was unsure about if (a) was a linear transformation. The other ...