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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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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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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2answers
18 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
20 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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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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13 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
14 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
27 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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38 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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3answers
32 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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2answers
33 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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2answers
27 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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0answers
20 views

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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2answers
27 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
16 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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1answer
30 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
31 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
21 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 ...
2
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0answers
37 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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0answers
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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
33 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
51 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
80 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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1answer
27 views

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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2answers
23 views

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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0answers
29 views

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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0answers
16 views

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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25 views

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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0answers
12 views

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 \\ ...
1
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2answers
38 views

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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0answers
21 views

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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0answers
18 views

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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19 views

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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0answers
56 views

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 ...
2
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4answers
52 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 ...
2
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1answer
36 views

Does $[V(\lambda)/W(\lambda)] = [V/W](\lambda)$?

Let $T$ be a linear operator on a vector space $V$, and let $W$ be an invariant subspace. If $V(\lambda)$ denotes the $\lambda$-eigenspace of $V$ and $W(\lambda)$ the eigenspace of $T$ on $W$, then ...
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6answers
176 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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1answer
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Several true/false statements about a finite group $a,g\in G$ such that $a$ is of order $2$

Let $G$ be a finite group, and $a,g\in G$ such that $a$ is of order $2$, then the following is either true or false: The element $gag^{-1}$ is of order $2$. $(ag)^2=g^2$ if $ag$ is of ...
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Find the dimensions spanned by the vectors [on hold]

Compute the dimension of the subspace spanned by each subset. $\{1, e^{ax}, xe^{ax}\}, \{1, \cos 2x,\sin^2 x\}$
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1answer
50 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 ...
3
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1answer
30 views

linear transformation proof problem

So question is : For any $m\times n$ matrix $A$,let $T_A$ be the linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ defined by $T_A(x) = Ax$ for all $x \in \mathbb{R}^n$.Let $A$ and $B$ be ...
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1answer
17 views

Country ranking by combination of factors [on hold]

I'm trying to find the most correct way of ranking countries based on multiple factors with measurements in different units. Take the following example: I am comparing $4$ countries nl.: United ...
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1answer
11 views

Basis for range and kernel of T and prove rank nullity theorem

$$ T \begin{bmatrix} x_1 \\ x_2 \\ \end{bmatrix} = \begin{bmatrix} x_1 +3x_2\\ x_2 \\ \end{bmatrix} $$ By considering $ker(T)$ first ...
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1answer
32 views

Degenerate quadratic form

I'm beggining with quadratics forms and I ma wondering : Let $a$ be a real number and $q:\mathbb{R^4} \rightarrow \mathbb{R}$ $(x,y,z,t) \rightarrow ax^2+2axy+y^2+4zt-at^2$ I would like to know for ...
3
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1answer
54 views

How to prove that for all $k\in\mathbb N$, $h(kx)=kh(x)$ and $h(x+y)\le h(x)+h(y)$?

Suppose $X$ is a commutative monoid and $f:X\to\mathbb R\cup\{\infty\}$ a function and $$g(x)=\inf\left\{\sum_{i=1}^nf(x_i)~\middle\vert~\sum_{i=1}^nx_i=x,n\in\mathbb N\right\}$$ ...
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1answer
32 views

Is this a linear transformation? in the context of group representations

Let $G$ be a group. A regular representation is given as $V=\mathbb{C}[G]$, a vector space, where $l: G \to GL(V)$ be the action is given by $l(g)(\alpha)(h) = \alpha (g^{-1}h)$ for all $g,h\in G, ...
0
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1answer
41 views

True/false questions about minimal and characteristic polynomials of a matrix

We have the matrix $A= \begin{pmatrix} 0 &2 &2 \\ 2& 0 &2 \\ 2& 2 & 0 \end{pmatrix}$, then one of the following is true: $f_A(x)=m_A(x) $ The matrix ...
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1answer
32 views

How to go about proving dim(VxW) = dim V + dim W [on hold]

I'm lost on how I should go about thinking and proving this.. Any help would be much appreciated.