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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1answer
149 views

Check if the following gradient is correct

This question regards the verification of the gradient of a given function. Notation. Let $N, K \in \mathbb{N}_0$ be given (nonzero) integers, with $K > N$. Let $\mathbf{x} = [x_b \ y_b \ z_b]^T ...
3
votes
1answer
31 views

Expressing components of a block matrix

Say you have a block matrix of $n \times n$ matrices $M = \begin{pmatrix} A&B\\ C&D \end{pmatrix}$ and you know rank ($A$) = rank($M$) = $n$. Show that $D= CA^{-1}B$. I'm really confused as to ...
1
vote
1answer
121 views

Proving minimum polynomial equals characteristic polynomial in a cyclic vector space

Let $V$ be a vector space over a field $\mathbb{F}$ and $T:V \rightarrow V$ be a linear map. Let $v \in V$ be such that $\left \{ v,T(v),T^{2}(v)... \right \}$ spans $V$. I have proved that $B=\left ...
0
votes
1answer
22 views

Local boundedness of linear operators

According to this definition of a bounded linear operator L(v): X -> Y the bounding constant M must be the same for all elements of the preimage X of the operator. However it then says that a bounded ...
3
votes
1answer
66 views

let $\lambda, \mu$ be distinct eigenvalues of a $2 \times 2 $ matrix $A$.Then which of the following statements must be true?

I was thinking about the following problem.. let $\lambda, \mu$ be distinct eigenvalues of a $2 \times 2 $ matrix $A$.Then which of the following statements must be true? a. $A^2$ has ...
1
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1answer
72 views

Find matrices complying to given constraints

We are given linear mapping of $n$-dimensional vector space, such as: It has $n+1$ eigenvectors Any $n$ of them are linearly independent Find all matrices which could define such a linear ...
3
votes
1answer
91 views

Find rank of the matrix $a_{ij}=(i-j)^2$, $i,j=1,\dots, n$

Let is $A$ $n\times n$ matrix defined in following way $a_{ij} = (i-j)^2$. For example when $n=4$ $$ A= \begin{pmatrix} 0&1&4&9\\ 1&0&1&4\\ 4&1&0&1\\ ...
3
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0answers
79 views

Sum of Gauss sum

Let $p$ be an odd prime, $v \in \mathbb{N}$ be a positive integer, and $c\in \mathbb{Z}$. Set \begin{align} G(c,p^v):=\sum_{\substack{d \bmod p^v \\ (d,p^v)=1}}{ \left(\frac{d}{p^v}\right) {e}^{ { ...
1
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1answer
35 views

Evaluate T* at the given vector in $C^2$

I have an Inner product space V, where $ V= C^2$, $ T(z_1,z_2) = (2z_1 + iz_2, (1-i)z_1), $ $ x= (3-i, 1+2i). $ I need to evaluate $T$* at the given vector x in V. So I did: $ \lt T( z_1,z_2) ...
0
votes
1answer
33 views

How to evaluate the accuracy for sparse linear system solver

I'm currently trying to do some experiments on linear solver. However, it's a little hard to get the sense of the numbers. For example, I know large condition number is bad, but how large is bad? ...
0
votes
1answer
62 views

properties of left-invertible matrix

When reading the notes on Left-invertible matrix, where $A$ is a matrix of dimension of $m\times n$, and $X A=I$. It is claimed that it $m$ must be larger than $n$, and $rank(A)=n$.How to get these ...
2
votes
1answer
42 views

Showing the existence of an eigenvalue whose real part is positive

$$M = \left(\begin{array}{cc|cc|cc|cc|cc} -b_1 &0 &b_2 &0 &0 &0 &\ldots &\ldots &0 &0\\ 0 &-a_1 &0 &a_2 &0 &0 &\ldots &\ldots &0 ...
0
votes
1answer
19 views

Symmetric matrix $S$ with collapsed eignevalues $\lambda_i=\alpha$ implies $S=\alpha I$.

Suppose we have a symmetric $n \times n$ matrix $S$ whose eigenvalues all collapse to a single eigenvalue $\alpha$. Show that $S = \alpha I$. From this it is clear that the characteristic equation ...
1
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2answers
962 views

Given two unit vectors, find a vector perpendicular with additional constraint

Given two unit length vectors find a perpendicular vector of unit length. I want to know if there's a way to do this without using a square root operation (avoid a normalization operation). Since the ...
1
vote
1answer
70 views

How do I rewrite vectors in other basis' given change of coordinate matrices?

$\displaystyle β= \begin{bmatrix}2\\2\\\end{bmatrix}$,$\displaystyle \begin{bmatrix}4\\-1\\\end{bmatrix}$ $\displaystyle C= \begin{bmatrix}1\\3\\\end{bmatrix}$,$\displaystyle ...
13
votes
1answer
237 views

Compute $\det(A^n+B^n)$

Let $A, B $ be two real $3\times 3 $ matrices, $AB=BA$, and $ \det(A-B)=\det(A^2+B^2)=1,\det(A+B)=3, \det(B)=0 $, then, what is ? $$\det(A^n+B^n)$$ here $n$ is a positive integer. The problem ...
3
votes
1answer
51 views

Diagonalisability…without the characteristic polynomial

Let us consider an $n\times n$ matrix $A$ defined as follows $$ A=\begin{pmatrix} 1+a&1&\cdots &1\\ 1&1+a&\ddots&\vdots\\ \vdots&\ddots&\ddots&1\\ ...
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0answers
60 views

Notation in Linear Algebra

What does $(A\mid b)$ denote in Linear Algebra? Specifically in the context of the following question: "If $(A\mid b)$ is in reduced row echelon form, prove that A is also in reduced row echelon ...
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3answers
93 views

How does this prove: ALL the eigenvalues of a triangular matrix = ALL of its diagonal entries? [Lay P269 Theorem 5.1.1]

For simplicity, consider the $3\times 3$ case. If $A$ is upper triangular, then $ A-\lambda I=\left\{\begin{array}{lll} a_{11} & a_{12} & a_{13}\\ 0 & a_{22} & a_{23}\\ 0 & 0 ...
0
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2answers
59 views

Solve Linear Sytem of Equation for $u,v,w$

I need to solve this sytem for $u,v,w$. I´ve tried basic algebra, but my answer does not mach the one from the book.
3
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0answers
38 views

The relationship between CPTP maps and quadratic forms

Let $H$ be a finite-dimensional Hilbert space (so there is a canonical isomorphism $H\cong H^*$). For a Hilbert space $H$ define $B(H)$ to be the space of linear operators on $H$; we have $B(H)\cong ...
0
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1answer
49 views

Jordan matrices and subspaces

I have the next problem: Find all the invariant subspaces over $\mathbb{R^4}$ of the following endomorphism given by the matrix A. The matrix A doesn't matter. What it does is the next: I found the ...
0
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1answer
43 views

Finding eigenvalues of a two-state system

Let $$A=\left[\begin{matrix} 2 & -i \\ i & 2 \end{matrix}\right],$$ Show that $U_1 = \dfrac{1}{\sqrt{2}}(\Psi_1+i\Psi_2)$ and $U_2 = \dfrac{1}{\sqrt{2}}(\Psi_1-i\Psi_2)$ are ...
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2answers
52 views

GCSE maths question concerning indices

Today was my GCSE maths exam, and I found it perfectly straightforward - except for one question. The question was: Given that $(2^\frac1 2)^n = \frac{2^x} {8^y}$ Find $n$ in terms of $x$ and $y$ ...
2
votes
2answers
156 views

Find a matrix such that $Ax=0$

Let $$W = span\left\{ {\left( {\matrix{ 1 \cr 0 \cr 0 \cr 1 \cr } } \right),\left( {\matrix{ 0 \cr 2 \cr 1 \cr { - 1} \cr } } \right)} \right\}$$ I was asked ...
2
votes
1answer
80 views

Finding a basis for intersection of two subspaces

Let two subspaces of $V=\mathbb{R}^4$: $$w1 = \left\{ {\left( {\matrix{ 1 \cr 1 \cr 1 \cr 1 \cr } } \right),\left( {\matrix{ 1 \cr 0 \cr 2 \cr 0 \cr } } ...
0
votes
1answer
48 views

Rank Nullity and Dimension relation

How would one prove the relations: $rank S◦T = rankT-dim(kerS ∩ ImT)$ and $nullity S◦T = nullityT+dim(kerS ∩ ImT)$ I understand that the use of rank nullity theorem is required but am confused by ...
1
vote
2answers
29 views

Why are the congruences $p^2-1 \equiv 0(\mod 8)$ and $p^e \equiv 1 + e(p-1) (\mod 4)$ for odd prime $p$ and $e \ge 1$ true?

Why are the congruences $p^2-1 \equiv 0(\mod 8)$ and $p^e \equiv 1 + e(p-1) (\mod 4)$ for odd prime $p$ and $e \ge 1$ true ? Suppose $p$ is an odd prime. I see easily that $p-1 \equiv 0 (\mod ...
0
votes
1answer
351 views

Finding the representing matrix with respect to the standard basis.

Let $B=[(1,0,0),(1,2,0),(1,2,3)]$ be the basis for $\mathbb{R}^3$. Let $T:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a linear transformation, such that its representing matrix with respect to basis $B$ ...
1
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4answers
81 views

Why is a set of orthonormal vectors linearly independent?

A set of orthonormal vectors is linearly independent. Since I wouldn't memorize any such facts that is mentioned in my course book, what is the reason behind it and why are they linearly ...
0
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0answers
72 views

distance between 2 parallel hyperplanes (non-trivial)

I am trying to solve the problem shown below in the image. And I was looking at the solution, I didn't know how the solution managed to come up with $$x_1 = (b_1 / || a ||^2) a $$ and $$x_2 = ...
1
vote
1answer
85 views

Factorise matrix determinant

How do I reduce this matrix to row echelon form and hence factorise the determinant, or is there a way that I am unaware of that factorise the determinant of this matrix without having to reduce it ...
1
vote
4answers
68 views

Find matrix determinant

How do I reduce this matrix to row echelon form and hence find the determinant, or is there a way that I am unaware of that finds the determinant of this matrix without having to reduce it row echelon ...
1
vote
1answer
45 views

finding polynomial in basis

Given a polynomial function p(t) = t^3 - t^2 + t - 4 in standard basis {1, x, x^2, x^3} Find the same polynomial function in a different basis B = {1, x-2, (x-2)^2, (x-2)^3} Currently im creating a ...
3
votes
2answers
140 views

About linear map and null space

Find a linear map $T:\mathbb{Q}^{6}\rightarrow \mathbb{Q}^{4}$ such that the kernel of T be the subspace of $\mathbb{Q}^{6}$consisting of all the vectors $(a_{1},...,a_{6})$ such that $\displaystyle ...
4
votes
1answer
95 views

Is there any proposition in real analysis or linear algebra that can only be proved by contradiction?

By "only be proved by contradiction", I mean either it's probable that this proposition can only be proved using contradiction, or that no one has ever came up with a direct proof. An undergraduate ...
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2answers
44 views

Constructing a matrix that computes derivatives

Consider the subset of functions given by $S = \text{Span}(e^{2t}\, \sin\, 3t, e^{2t}\, \cos\, 3t)$: Show that the derivatives of $e^{2t}\, \sin\, 3t$ and $e^{2t}\, \cos\, 3t$ are also in $S$ and ...
0
votes
2answers
221 views

Weird matrix row reduction to row echelon form to find determinant

How do I reduce this matrix to row echelon form and hence find the determinant, or is there a way that I am unaware of that finds the determinant of this matrix without having to reduce it row echelon ...
2
votes
1answer
99 views

Having trouble considering a finite field $\mathbb{F}_{p^n}$ as a vector space over $\mathbb{F}_p$.

As the title states, I'm having trouble considering a finite field $\mathbb{F}_{p^n}$ as a vector space V over $\mathbb{F}_p$. Clearly it is dimension $n$. How can we work with this vector space? ...
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1answer
126 views

Real symmetric $\;2\times 2$ congruent matrices

Theorem: Prove that any two regular symmetric $\;2\times 2\;$ real matrices are either simultaneously diagonalizable or else they are congruent to each other. Now, my work: If $\;A,B\in Sym_2(\Bbb ...
0
votes
1answer
46 views

Every Functional on V has the form T(x) = <x,y> for a unique y in V?

Every functional on V has the form T(x) = for some unique vector y in v? V is a finite-dimensional space. The book says it's false, but it is the book that is wrong. The end.
1
vote
1answer
55 views

Vector equation of planes

So I have these two planes: $$\pi_1: X = (1,0,0) + \lambda(0,1,1) + \gamma(1,2,1)$$ $$\pi_2: X = (0,0,0) + \lambda(0,3,0) + \gamma(-2,-1,-1)$$ I need to find two points $A$ and $B$ of the ...
0
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1answer
41 views

question related to linear algebra

Let $V$ be a finite dimensional real vector space and let $f$ and $g$ be non- zero linear functionals on $V$. Assume that $ker(f)$ is contained in $ker(g)$. Which of the following statements are ...
0
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1answer
204 views

Regular matrix and regular stochastic matrix

We know that : A matrix is regular if its determinant is non zero. A stochastic matrix is regular if at a certain power all elements are positive. Question is how can I make the link between the ...
0
votes
2answers
24 views

Find the standard matrix for the linear transformation T?

T is the clockwise rotation ($\alpha$ is negative) of 30$^o$ in $R^2$, v=(2,1). Then find the image of the vector v. I'm bit confused with this problem as I tried with different methods even ...
3
votes
2answers
200 views

Eigenvectors of the Zero Matrix

Given the following matrix: $ \begin{pmatrix} -1 & 0 \\ 0 & -1 \\ \end{pmatrix} $. I have to calculate the eigenvalues and eigenvectors for this matrix, and I have calculated that ...
0
votes
2answers
69 views

Why are these vectors expressed as row vectors and not column vectors? When to write as row vectors or column vectors?

Everytime I have been asked to find a basis when the vectors were given in comma delimited form, I, and the book, would write out the vectors as columns in a matrix. Another example in the book ...
0
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1answer
44 views

find minimal polynomial of $T(p)=p'+p$

I'm trying to solve the following question: let $T: \mathbb C_n[x] \to \mathbb C_n[x]$, $T(p)=p'+p$ find the characteristic and minimal polynomial of $T$. What I'm trying to do is the following: I ...
2
votes
0answers
64 views

The canonical perspective on the Hodge star operator [closed]

I am looking for the canonical perspective on the Hodge star operator. I want to see it done properly, not using basis for its definition, saying clearly what we assume in its definition. ...
0
votes
1answer
41 views

Is this rank $1$ matrix is semidefinite?

I have a matrix, $X = xx^T$, where $x \in \mathbb{R}^n$. Is the matrix $X$ semidefinite?