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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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 $X$ ...
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1answer
24 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 ...
6
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0answers
54 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 ...
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1answer
17 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 ...
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1answer
8 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
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2answers
54 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
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1answer
29 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 ...
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2answers
15 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 = ...
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3answers
19 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
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1answer
35 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. ...
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2answers
68 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 ...
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3answers
33 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
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1answer
35 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
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1answer
70 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 ...
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0answers
14 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
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1answer
24 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 ...
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1answer
31 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 ...
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1answer
27 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 ...
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2answers
39 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 ...
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2answers
40 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 ...
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0answers
27 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 ...
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0answers
28 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 ...
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2answers
27 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: ...
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3answers
183 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 $\,p(t)=t^3-t+1$, and $A$ is invertible since ...
0
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1answer
28 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 ...
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1answer
26 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.
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2answers
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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
46 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 ...
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1answer
30 views

Let $V$ be a one dimensional vector space.Find all linear maps from $V$ to $V$.

Let $V$ be a one dimensional vector space over a field $F$.Find all linear maps from $V$ to $V$. My try: Let $\{x_0\}$ be a basis of $V$ .Then any $x\in V$ can be expressed as $x=cx_0;c\in F$. Now ...
2
votes
2answers
30 views

Solution Space - Linear Algebra

For a matrix:\begin{bmatrix}-1&2&3&-3&6&7\\ 1&-1&-2&2&-5&-6\\ -1&1&2&-1&2&4\\ -2&2&4&-2&4&8\\\end{bmatrix} To solve for ...
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1answer
39 views

When does $ \langle gI, t \rangle = \langle I, g^{-1} t\rangle $ hold true?

Consider $I, t \in \mathbb{R}^d$ and $g$ is some element in a group of transformations (for example like the affine group in $\mathbb{R}^2$). I was wondering when the inner product $ \langle gI, t ...
0
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1answer
43 views

Least squares and pseudo-inverse

Let $b\in \mathbb{R}^m$,$A\in M_{m\times n}(\mathbb{R})$ with $m>n$ and $rank(A)=n$, and the element $x^*\in \mathbb{R}^m$ solution of least squares of $Ax=b$. i) Show that $r^*=b-Ax^*\in N(A^T)$ ...
0
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1answer
26 views

Projection on the coordinate plane

Consider the vector space $\mathbb{R}^3$ with usual inner product. Find the orthogonal projection matrix on coordinate plane $xy$ and $xz$ I think that projection on xy is ...
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0answers
20 views

Linear Algebra and Upper triangular matrix [on hold]

What are the commutation brackets for bases of group of upper triangular 2*2 matrices with three different components?I know the bases but I don't know what brackets have resulted them.
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2answers
81 views

Orthogonal projection matrix

Let $A\in M_{m\times n}(\mathbb{R})$. Denoting by $R(A)$ the column space of $A$ and $N(A)$ the null space of $A$. I know that $z^*=Ax^*$ is a projection of $b\in R^m$ on $R(A)=N(A^T)$ where ...
0
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4answers
87 views

Is an non-invertable matrix an linear operator?

I am under the impression that any matrix can be called a linear operator, even if the matrix does not have an inverse. Is it true? There are many properties a linear operator enjoys; do all matrices ...
0
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1answer
18 views

Effect of spectral shift on the eigenvalues of a real symmetric matrix [duplicate]

Suppose a matrix A(real symmetric) is changed to A − σ I, where σ is any scalar quantity and I is the identity matrix. Explain what happens to the eigenvalues and eigenvectors of A? I am unable to ...
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1answer
38 views

Connection between invertible matrix and eigenvalues

In many questions I read lately about eigenvalues I am asked to proove theorms about the connection between invertible matrices and their eigenvalues. Given an invertible matrix $A\in ...
0
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1answer
23 views

Length of projection onto a subspace equal length of the vector

If the length of the projection of a vector onto a subspace equals the length of the vector, does this always imply that the vector belongs to that subspace. This was quite easy to show when the ...
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2answers
31 views

Enough Information? (Linear Algebra over Finite Fields)

This problem works over the field $\mathbb{F}_p$. Suppose $p$ is a prime and the $i\in I$ index the set of $p$ vectors $v_i$. Fix $k$ an integer with $1\leq k<p$. Let $v_i$ have the following ...
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1answer
50 views

prove that a matrix only have positive eigenvalues

A is a $N\times N$ matrix with all positive elements and only has positive eigenvalues. (If $A$ has negative elements please refer to the first answer for an counter-example). $B$ is $N\times 1$ ...
2
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1answer
25 views

On the expression of the Galois conjugates in terms of the coordinates in a basis

Let $K$ be a field and let $L$ be a Galois extension of $K$. Assume that $[L:K]=n$, and consider $e=(e_1, e_2, ...,e_n)$ a basis of $L$ over $K$. We note ...
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1answer
28 views

Linear algebra: Dimension of column space

Let $A$ be an $n\times n$ matrix. Dimension of its null space is $m < n$. What is the dimension of its column space? Can some one please explicitly explain this? What I am majorly confused ...
0
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1answer
30 views

Linear Algebra - Change of basis

Let $S$ be the standard basis for $\mathbb{R}^5$. Let $B = (b_1, b_2, b_3, b_4, b_5)$ be the ordered basis with: $b_1 = (2, 1, 1, -2, -2)$; $b_2 = (0, -2, 4, 5, -4)$; $b_3 = (1, -4, 5, 5, -4)$; ...
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1answer
33 views

Minimum matching convolution

Let $\text{SPD}^n$ and $\text{PD}^n$ be the semi-positive and positive definite matrices in $\mathbb{R}^{n\times n}$, respectively. I want to find an $X\in \textrm{SPD}^n$ that minimizes $||X||$ ...
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1answer
34 views

Find a matrix whose column space contains the column space of the given matrix.

Let $$A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}\text{.}$$ $C(A)$ denotes the column ...
0
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1answer
17 views

Linear functional and Hessian

Consider the vector space $\mathbb{R}^n$ provided with the usual inner product $<.,.>$. Let $A\in \mathbb{M}_n(\mathbb{R})$ a invertible matrix, $b\in\mathbb{R}^n$ and $J:\mathbb{R}^n\rightarrow ...
0
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1answer
38 views

Do the spaces spanned by the columns of a matrix and by the columns of a set of matrices coincide?

As in Do the spaces spanned by the columns of the given matrices coincide?, let $$A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ ...
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1answer
77 views

How to show the following matrix is positive definite?

How to show the following matrix is positive definite. \begin{equation} \sum_{i=1}^n \Big[(d_i^Tp)^2\left\{\left( \begin{array}{c} d_i\\ A_ip \end{array} \right) \left( \begin{array}{c} ...
0
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1answer
26 views

Uncountable “relatively independent” subset of finite dimensional vector spaces over an uncountable field

Let $V$ be a $n$ dimensional vector space over an uncountable field ; then does there always exist an uncountable subset $S$ of $V$ such that any $n$ vectors of $S$ are linearly independent ? ( I can ...