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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Linearity and invertibility of $A^{-1}$

If $A\in L(X)$ then prove that $A^{-1}$ is linear and invertible. Proof: Since $A$ is invertible then $A$ is injective and surjective. We know that $A^{-1}$ defines by $A^{-1}(Ax)=x$. Remark: Also ...
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14 views

Symmetric Matrix with Positive Eigenvalues

Not all matrix with positive eigenvalues is positive definite, i.e. $\mathbf{x}^\mathsf{T}A\mathbf{x}>0$ for all non zero vector $\mathbf{x}$. For example consider matrix $$A = \begin{bmatrix} 1 ...
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27 views

Show that $B^TAB$ is symmetric. [on hold]

$A$ is invertible, but it does not say that $A$ is symmetric. By $B^T$ I mean that $B$ is transposed.
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21 views

Is the the statement is true or false?

Suppose $A$ is a $m \times n$ matrix and $V$ is a $m \times 1$ matrix with both $A$ and $V$ having rational entries and suppose the system $AX=V$ has a solution in $\mathbb{R}^n$. Then the equation ...
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1answer
13 views

If $(A-\lambda I)^{k_j} \vec{v_j} = \vec{0}$ then $(A-\lambda I)\vec{v_j} = V_j$ and $V_j\in \ker(A-\lambda I)^{k_j-1}$

In one book on differential equations and dynamical systems I read that if (1) $(A-\lambda I)^{k_j} \vec{v_j} = \vec{0}$ then (2) $(A-\lambda I)\vec{v_j} = V_j$ and $V_j\in \ker(A-\lambda I)^{k_j-1}$. ...
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1answer
27 views

Path to Self Study Calculus 1-4 and Linear Algebra [on hold]

For the past year I've taken up self studying mathematics. My initial intent was to study so that when I entered college (currently a junior) I would have most of the basic mathematics for studying ...
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1answer
19 views

$U,W$ are subspaces. show $\dim(U+W) = 1+\dim(U \cap W)$, then $\{U+W,U\cap W\}=\{U,W\}$

This is a question from a review package that is causing me some trouble. Let $U,W$ be subspaces of a finite dimensional vector space. Show if $\dim(U+W) = 1+\dim(U \cap W)$, then $\{U+W,U\cap ...
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1answer
19 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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2answers
31 views

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 ref are in the first and second columns). So the ...
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28 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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13 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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1answer
21 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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1answer
13 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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42 views

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
14 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
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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21 views

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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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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29 views

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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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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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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6 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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19 views

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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2answers
32 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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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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1answer
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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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2answers
26 views

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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24 views

$A$ 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
29 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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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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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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1answer
18 views

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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1answer
20 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
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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48 views

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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34 views

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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35 views

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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28 views

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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How to find the Cartesian equation of a plane in this example (in details)?

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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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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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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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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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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1answer
36 views

Producing a $G$-invariant form from the standard Hermitian product using the averaging process

Problem statement: Let $G$ be a cyclic group of order $3$. The matrix $$A=\begin{bmatrix} -1 && -1 \\ 1 && 0 \end{bmatrix}$$ has order $3$, so it defines a matrix representation of ...
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1answer
3 views

Determinant of block matrix with off diagonals as vectors

I've been given an $(n+1)\times(n+1)$ square matrix, which is written in the form of a block matrix with the following dimensions $ \begin{bmatrix} (1\times1) & (n\times1)\\ ...
2
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39 views

Linear transformation as dot product

Prove that to every $A\in L(\mathbb{R}^n,\mathbb{R}^1)$ corresponds a unique $\mathbf{y}\in \mathbb{R}^n$ such that $A\mathbf{x}=\mathbf{x}\cdot \mathbf{y}$. Prove also that $\Vert A ...
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22 views

Linear Algebra Vector Space and Subspace [on hold]

If $X$ be an infinite dimensional vector space and $Y$ is subspace of $X$, then show that whether dimension of $Y$ is always finite or infinite also. Also give example of any subspace whose dimension ...
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22 views

Why does $\bar A = \left\{ {{P_\Delta }(\lambda ):\left\| {{\Delta _j}} \right\| \le \varepsilon ,j = 0,1,2…m} \right\}$? [on hold]

Suppose ${P_\Delta }(\lambda ) = ({A_m} + {\Delta _m}){\lambda ^m} + ....... + ({A_1} + {\Delta _1}){\lambda ^1} + ({A_0} + {\Delta _0})$ is a matrix polynomial, and $\lambda $ is a complex ...