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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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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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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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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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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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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How to determine that the 3 points given in homogeneous coordinates are collinear?

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
17 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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11 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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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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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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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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20 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

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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24 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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$A$ 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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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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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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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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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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17 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
30 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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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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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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34 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)? [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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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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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
35 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
25 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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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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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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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 ...
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1answer
34 views

Completeness of bounded linear maps

Let $X,Y$ be normed vector spaces over $\mathbb{C}$, and $L(X,Y)$ the space of all bounded linear maps from $X$ to $Y$. Its known that $L(X,Y)$ is a normed(operator norm) vector space. Theorem: ...
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3answers
53 views

What is the meaning of $ \mathbb{R}^n$ to $\mathbb{R}^{n+1}$? [on hold]

In linear algebra, what does it mean to go from $\mathbb{R}^1$ to $\mathbb{R}^2$ or $\mathbb{R}^2$ to $\mathbb{R}^3$?
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4answers
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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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1answer
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Matrix multiplication to make all numbers in a 3x3 matrix negative

Let's say I have the matrix called Delta, $$ \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{matrix} $$ What would I have ...
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Operations on 3x3 matrix through matrix products

What would I have to multiply the following matrix ... $$ \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{matrix} $$ by so ...
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A fibre bundle of n-tuples of vectors

The question is: does the surjective, multilinear map $\pi: (\mathbb R^\infty)^n\to \Lambda^n(\mathbb R^\infty)$ defined by $(v_1,\cdots,v_n)\mapsto v_1\wedge \cdots \wedge v_n,$ define a fibre ...
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Matrix shear transformations [on hold]

If you know the line of a shear transformation (the invariant line), how word you go about finding the shear factor? Also, funny as it may sound, what is the shear factor - what does it show?
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how to know if linear combinations fill a line, plane, or $R^3$?

I just started taking linear algebra and I am already confused about how to know whether linear combinations fill a line, plane, or $R^3$, my textbook simply says it you have one vector with a scalar ...
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Reduce a matrix to row-echelon form with partial pivoting

Use the Gaussian elimination with partial pivoting manually to reduce the following matrix to row echelon form: $$ \begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ ...
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Showing that if $A$ is diagonalizable then $A^2-4A+8I$ is diagonalizable

Let $A_{n\times n}$ be a real matrix then: if $A^4 = 8A$ then $A$ is not invertible. if $A$ is diagonalizable over $\mathbb R$ then $A^2-4A+8I$ is diagonalizable over $\mathbb R$. ...
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Uniqueness of spectral decomposition

Suppose $T: V\rightarrow V$ is diagonalizable on an arbitrary vector space (not necessarily an inner product space), so $T = \sum_{i=1}^r\lambda_i P_{\lambda_i}$ where ...
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Projection of a Vector on a Straight Line in $\mathbb{R}^3$

I have the following: Consider the straight line $(\epsilon)$ which passes through the origin and forms an angle $t$ with $Ox$ axis. Find the matrix $A$ which projects a random vector ...
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question about gcd

I encountered the following question: Find a natural number x which satisfies the following: 12345 mod 54321 = 6 mod 54321 I tried using the extended Euclidean algorithm, but failed to solve the ...
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A method of writing all primes

I've recently noticed a method of describing primes. As an example: $13=5*11-2*3*7$. This pattern must follow these rules: $x-y$ such that $x*y$ is the product of all previous primes (allowing powers ...
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4answers
57 views

How does computing the determinant of a matrix with unit vectors give you the Cross Product?

Say you had $(a_x,a_y,a_z)\times(b_x,b_y,b_z)$, you would set up a matrix like the following: And the resulting would be your Cross Product or the coordinates of an orthogonal vector. My question ...