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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Finding a Matrix B by knowing its Kernel is the Image of Matrix A

I understand how to find the image(A). The basis of Im(A) would be the first two columns of the matrix A (given the two leading 1's in rref are in the first and second columns). So the Ker(B) = ...
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1answer
16 views

Finding an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$

Problem Statement: Let $A=\begin{bmatrix} 2 && 1 \\ 1 && 2 \end{bmatrix}$. Find an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$. I am ...
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1answer
43 views

Finding the inverse of A where A is of the form $A = D (I − N)$, where $D$ is diagonal with nonzero entries and $N$ is nilpotent

If a matrix can be written as $A = D (I − N)$, where $D$ is diagonal with nonzero entries and $N$ is nilpotent, then $A^{−1} = (I − N)^{−1}D^{−1}$. Use this to find inverse of: $\begin{bmatrix} 2 ...
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28 views

symplectic base in $\mathbb{R}^{2n}$

Please, can somebody help me? In the vectorial space $\mathbb{R}^{2n}$,is the canonical basis a symplectic one?
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27 views

Understanding a basic matrix theorem

There's a theorem in Linear Algebra which says that if ${\bf A}$ is an $m \times n$ matrix and $m < n$, then the homogeneous system of linear equations ${\bf A}{\bf X}=0$ has a non trivial ...
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12 views

Showing that span$\{x,Ix\}$ is an invarient subspace of $V:=\mathbb{R}^n$

Let $V := R^n$ be a vector space and let $I \in O(n)$ be an operator satisfying $I^2 = -Id$. I want to show that the $span\{x,Ix\}$ is an invarient subspace of $I$. Let $W = span\{x,Ix\}$. I need to ...
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Determinant from Paul Garret's Definition of the Characteristic Polynomial.

$\DeclareMathOperator{\id}{id} \DeclareMathOperator{\End}{End}$ On pg. 390 of Paul Garret's notes on Algebra, a definition for the characteristic polynomial is given, which I discuss here. Let $V$ be ...
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49 views

Cayley-Hamilton Theorem - Trace of Exterior Power Form

Let $V$ be an $n$-dimensional vector space over a field $F$ (the characteristic of which, for the purpose of this post, may be taken as $0$). Let $T$ be a linear operator on $V$ and $\lambda\in F$. ...
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1answer
58 views

Trace of the $k$-th Exterior Power of a Linear Operator

Let $V$ be an $n$ dimensional vector space over a field $F$ and $T$ be a linear operator over $V$. Assume that the characteristic of $F$ is not $2$. Definition. Consider the map $f_1:V^n\to ...
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19 views

Proving a basis spans $R^3$

Doing some reviewing and I'm not 100% sure if my thought-process is correct. I have the following two vectors and need to prove they're a basis for $R^3$: $$B= \begin{bmatrix} 1 \\ ...
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13 views

Least Squares Solution and Singular Vector

Is there a simple way to show that the least square solution of an overdetermined linear system is equal to the right singular vector of the coefficient matrix corresponding to the smallest singular ...
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1answer
11 views

Let $\{e_1,\ldots,e_n\}$ be an arbitrary basis in a finite dimensional inner product space. Prove $\exists \{f_1,\ldots,f_n\}: (e_i,f_j)=\delta_{ij}$

Let $\{e_1,\ldots,e_n\}$ be an arbitrary basis in a finite dimensional inner product space $V$. Prove there exists vectors $\{f_1,\ldots,f_n\}$ such that $(e_i,f_j)=\delta_{ij}$. I tried using ...
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1answer
39 views

isomorphic linear spaces [on hold]

Let $S$ be the space of $3\times 3$ skew-symmetric real matrices. Then $dim\ S=3$. Is it true that $S$ is a vector isomorphic to $\mathbb R^3$? What is the isomorphism then?
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Is $\{(x,y,z) \in \mathbb{R}^3 : x^2+3y^2+12z^2=0\}$ a vector space?

Is $\{(x,y,z) \in \mathbb{R}^3 :x^2+3y^2+12z^2 = 0\}$ a vector space? My inclination is that the only real solution to $x^2+3y^2+12z^2=0$ is $(0,0,0)$, which is the trivial subspace of ...
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1answer
15 views

How can we know if the minimal polynomial of a matrix has no multiple products?

If the characteristic polynomial $f_A(x)$ has multiples of the same product, for example $f_A(x)= (x+2)^2(x-1)$ so $(x+2)$ has a multiple of $2$, then is there a condition on $A$ such that we know ...
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Characteristic polynomial of $A^2$, given the characteristic polynomial of $A$

Let $A \in M^{\mathbb{R}}_{3x3}$, it's characteristic polynomial is $P_A(t) = t^3+t^2+t-3$. find the coefficient of the characteristic polynomial of $A^2$. I tried to solve it by finding the factors ...
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Why is the standard inner product on F^n equal to this?

In the textbook that I'm using the standard inner product is defined as $$\langle x,y\rangle = \sum_{i=1}^{n}a_{i}\overline{b_{i}}$$ where $x=(a_{1}, a_{2}, {...}, a_{n})$ and $y=(b_{1}, b_{2}, ...
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1answer
11 views

What is the dimension of $c_0/M$?

Let $c_0=\{ (x_n) : x_n\in \Bbb{R}, x_n \to 0\}$ and $M=\{(x_n)\in c_0 : x_1+x_2+\cdots + x_{10}=0\}$. Then, what is dim($c_0/M$) ?
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How to determine that the 3 points given in homogeneous coordinates are collinear? [on hold]

How do I prove that the 3 points given in homogeneous coordinates are collinear? $$A=(1,3,2)^T, B=(0,6,8)^T, C=(3,3,-2)^T$$
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1answer
24 views

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar?

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar? So I read that it's true only if $A,B$ are diagonalizable, but why? if the characteristic polynomial is ...
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0answers
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Set of conjugate vectors that span both Krylov space

If $P$ contains a set of conjugate vectors that span Krylov space of matrix $A$, i.e. $\mathcal{K}(A, x)$, and also $P$ span Krylov space of matrix $\mathcal{K}(B, x)$, is it true that the diagonal ...
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278 views

Centralizer of a Matrix over a Finite Field

Let $\mathbb F$ be a finite field. Denote by $M_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$. For a matrix $A\in M_n(\mathbb F)$ what is the cardinality of $C_{ M_n(\mathbb ...
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What is the maximum value of coefficient $f_v$ with the constraInt that the matrix is positive semi-definite?

I am trying to solve this equation my self with my knowledge about characteristic polynomials, etc but I have placed it here earlier because I'm not a mathematician and maybe you give me ideas to ...
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1answer
20 views

Sparse Matrix or Dense Matrix

My task is to implement the inner product and vector triad forms for a dense $A$ in single and double precision. I have successfully implemented the inner product and vector triad form although, I am ...
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Product of symmetric positive semidefinite matrices is positive definite?

I see on wikipedia that the product of two symmetric positive definite matrices is also positive definite. Does the same result hold for the product of two positive semidefinite matrices? My proof of ...
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0answers
14 views

PGL_n transformation fixing a set of n+2 points in general position must be identity

I was wondering if anybody can think of a slick and short argument to show that any $PGL_n(k)$ transformation $A$ that fixes a set of $n+2$ points $p_1, p_2,\dots, p_{n+2}$ in general position must be ...
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5 views

On a hypothetical computer with a word length of three digits and truncation, compute the solution of a system of equations

On a hypothetical computer with a word length of three digits and truncation, compute the solution of $$ \begin{matrix} -3x & + & y & = & -2 \\ 10x ...
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4answers
84 views

Show if $A^TA = I$ and $\det A = 1$ , then $A$ is a rotational matrix

Show if $A^TA = I$ and $\det A = 1$ where $ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $, then $A =\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & ...
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31 views

Find the matrix $P$ that multiplies $(x, y, z)$ to give $(y, z, x)$. Find the matrix $Q$ that multiplies $(y, z, x)$ to bring back $(x, y, z)$.

How do I solve these types of problems? What method or technique do I use? I want to learn how to solve the first one so that I can try to figure out the second part to the question myself.
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1answer
19 views

Find the solution set to the corresponding homogeneous system of equations

You are given a system of equations: $2w+3x-2y+z=-1$ $6w+10x+6z=14$ $3w+2.5x-15y-4.5z=-35.5$ and a particular solution to that system of equations, $\begin{bmatrix}0\\2\\3\\-1\end{bmatrix}$ ...
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1answer
10 views

is banded system the same with banded matrix in linear algebra [on hold]

I want to use SPIKE Algorithm to work out my parallel computing home work, but I am new to SPIKE Algorithm and I know nothing about Banded System Solver, I just ...
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1answer
28 views

Invariant subspace

Let $T: V \to V$ linear transformation, and let $W$ to be an invariant subspace of $V$. we mark $T_w: W \to W$ the from $T$ to $W$. Prove that if T is diagonalizable, then $T_w$ is diagonalizable. ...
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2answers
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$A$ is diagonalizable, if $A,B$ have then same eigenvalues, then $B$ is also diagonalizable

Given $A_{n\times n},B_{n\times n} \in \mathbb R$ such that $A$ is diagonalizable then: if $A,B$ have then same eigenvalues, then $B$ is also diagonalizable over $ \mathbb R$. if $A,B$ ...
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1answer
13 views

Why does $\text{dim}\,K^n = n$ for finite $n$ imply $\text{Im}(A^i)=\text{Im}(A^{i+1})$ for some $i\leq n.$

I'm studying about linear algebra and came across with the following: Let $A\in \mathcal{M}_{n\times n}(K)$ for some field $K$. If $\text{dim}\,K^n = n$ is finite then ...
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1answer
54 views

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$ …a different approach

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$. I have done the proof in a easy way… If $ABv = λv$, then $B Aw = λw$, where $w = B v$. Thus, as long as $w \neq 0$, it is ...
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Making a Matrix singular

During my research I came across the following problem. Intuitively this should be an easy one. However, the simplest version of it looks like this: Let $C \geq \frac{1}{2}$ be some fixed ...
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23 views

Twisted centralizer

Let $F$ denote a finite field and $A$ a square matrix with coefficients in $F$. The set of all matrices $B$ such that $BA=AB$ is called the centraliser of $A$. Now consider the set $C(a,A)$ of all ...
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Representing a linear operator on $V$ with an element of $V \otimes V^*$

I got interested by the first sentence of this wikipedia subsection. It claims that any linear operator $f:V\to V$ can be represented by an element of $V\otimes V^*$ in a very concrete way: the ...
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Let $A$ be a real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ?

Let $A$ be a square real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ? I know that real symmetric matrices are diagonalizable . Also if all the diagonal entries be ...
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1answer
31 views

Calculate A^8 using Cayley Hamilton Therorem

Find $A^8$ using Cayley Hamilton Therorem, when $$A = \begin{pmatrix} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 0 ...
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1answer
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If $A^4=4A^2$ then $m_A(x)=x^2-4$ and if it isn't diagonalaziable over $\mathbb R$ then $0$ is an eigenvalue

Given $A_{n\times n} \in \mathbb R$ such that $A^4=4A^2$ then if $A$ is invertible and isn't of the form $cI, c\in \mathbb R$ then $m_A(x)=x^2-4$. if $A$ isn't diagonalaziable over ...
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How to find the Cartesian equation of a plane in this example (in details)? [on hold]

I'm solving an A Level paper, and came across this question. Basically, they have given plane $p$ has the equation $(\mathbf r-3\mathbf i)\cdot(2\mathbf i-3\mathbf j+6\mathbf k)=0$. Now, I can see ...
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179 views

Can we say that $\det(A+B) = \det(A) + \det(B) +\operatorname{tr}(A) \operatorname{tr}(B) - \operatorname{tr}(AB)$.

Let $A,B \in M_n$. Is this formula true? $$\det(A+B) = \det(A) + \det(B) + \operatorname{tr}(A) \operatorname{tr}(B) - \operatorname{tr}(AB).$$
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Given $A$, $A^{-1}$ can be expressed with: $A^{-1}=bA+dI$

Given the matrix $A=\begin{pmatrix} -1 &3 &3 \\ 3& -1 & 3\\ 3& 3 & -1 \end{pmatrix}$ then $A$ is invertible and $A^{-1}$ can be expressed with: $A^{-1}=bA+dI, ...
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2answers
31 views

Find bases of the kernel and image

Find the rank and the nullity of the following linear map $T : U \to V$ , and find bases of the kernel and image of $T$. $U = \Bbb R^4 , V = \Bbb R^4$, $$T(α, β, γ, δ) = (α − γ, γ − δ, α − β, β − ...
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1answer
32 views

Given a normal $A_{n\times n}$ matrix, then $\lVert A^*v \rVert = \lVert Av\rVert$ and $\langle Av,v\rangle = \langle A^*v,v\rangle$

Let a normal $A_{n\times n}\in \mathbb C^n $ matrix, then: $\forall v \in \mathbb C^n:\lVert A^*v \rVert = \lVert Av\rVert $ $\forall v \in \mathbb C^n : \langle Av,v\rangle = \langle ...
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2answers
354 views

For what values of k and h does this system of equations have a unique solution?

Here's my system of equations: $x−3y+2z=5$ $2x−5y−3z=9$ $−x−y+kz=h$ So I have $ \begin{bmatrix} 1 & -3 & 2 & 5 \\\\ 2 & -5 & -3 & 9 \\\\ -1 & -1 & k& h ...
7
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1answer
152 views

Proving positive definiteness of matrix $a_{ij}=\frac{2x_ix_j}{x_i + x_j}$

I'm trying to prove that the matrix with entries $\left\{\frac{2x_ix_j}{x_i + x_j}\right\}_{ij}$ is positive definite for all n, where n is the number of rows/columns. I was able to prove it for the ...
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1answer
17 views

Which of the following statements is true?

(Q) is false since unitary matrix has modulus 1 eigenvalues. I think (P) is true but I am not sure how to Prove or Disprove this. Please suggest?
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3answers
52 views

linear algebra ( infinite dimensional space)

Does someone have a reference (book) in linear algebra on infinite dimensional space ? I don't know anything in French Literature. [Edit] I am sorry, i wasn't precise enough. I look for an advanced ...