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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16 views

Cramer's rule doesn't work here?

i tried to solve the following system: $$A_2*2isin( \beta a) = B_3*exp(- \alpha a)$$ $$i \beta A_2*cos( \beta a)2cos( \beta a) = - \alpha B_3*exp(- \alpha a)$$ then i got $A_2=0 \Rightarrow B_3=0$ ...
-3
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0answers
11 views

How do I set the vector $\vec{v} = (0, 0, 1, 0, 0)^T$ as $\vec{\vec{v}}=\vec{u}+\vec{w}$ with $\vec{u}\in U$ and $ \vec{w}\in U^\perp$?

Let U be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. enter image description here
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1answer
14 views

Let $V$ = $R^3$ and let $U$ be the subspace spanned by $A$= {$(-3,-2,0),(4,-1,2)$}. Is there a subspace $W$ of $V$ such that the following holds?

Let $V$ = $R^3$ and let $U$ be the subspace spanned by $A$= {$(-3,-2,0),(4,-1,2)$}. Is there a subspace $W$ of $V$ such that the following holds? $$W \nsubseteq U$$ $$dim(U)+dim(W)<dim(V)$$ ...
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2answers
16 views

How do I find a base of orthogonal complement $U^\perp$ of $U$ and determine the dimension of $U^\perp$?

Let U be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. How do I find a base of orthogonal complement $U^\perp$ of $U$ and determine the dimension ...
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1answer
21 views

If $Ax = O$ has only one solutions, then the columns of A: ${v1, v2…,vn}$ span R?

I've been doing some excersices about inner product and I found something interesting but I don't know if my approach is correct at all. Supose that ${v_{1}, v_{2}, ..., v_{n}}$ is a base for a ...
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4answers
36 views

How do I find orthonormal basis of $U$?

Let $U$ be the subspace of $\mathbb{R}^5$, which is through $(1,2,3,-1,2)^T$ and $(1,0,-1,0,1)^T$ spanned. How do I find orthonormal basis of $U$?
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4answers
58 views

Finding a basis for the set of polynomials where f(1)=f(-1)=0

I have the vector space $V$ above that belongs to $\mathbb{F}$, and $V$ is the group of all polynomials that are of degree $3$. $W= \{ p \in V | p(1)=p(-1)=0\}$ 1.) Prove that W is a subspace of ...
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2answers
22 views

The determinant of the transposing endomorphism

Let $K$ be a field and $f$ the endomorphism of $\mathcal M_n(K)$ that sends a matrix to its transpose. I want to determine the determinant of $f$. I know that since $f^2=id$ then $det(f)=1\ or \ -1 $ ...
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11 views

Can we exploit FFT for evaluating quadratic on gridded data with stationary covariance?

I would like to evaluate the quadratic $\mathbf{y}^{T}K^{-1}\mathbf{y}$ with the following assumptions: The entries of $\mathbf{y}$ are $y_i = f(\mathbf{x_i})$ which correspond to points on a ...
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1answer
14 views

Show that the operator induced by $T$ on the quotient space $V/\operatorname{ker} (T-5I)$

A linear operator $T$ on a complex vector space $V$ has characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^2(x-5)$. Show that the operator induced by $T$ on the quotient space ...
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2answers
36 views

Is the set of periodic functions from $\mathbb{R}$ to $\mathbb{R}$ a subspace of $\mathbb{R}^\mathbb{R}$?

A function $f: \mathbb{R} \to \mathbb{R}$ is called periodic if there exists a positive number $p$ such that $f(x) = f(x + p)$ for all $x \in \mathbb{R}$. Is the set of periodic functions from ...
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22 views

Suppose A and B are two matrices so that AB=0, which are true, if any, why? [on hold]

$ col(A)\subseteq null(B)$ $ null(A^TB^T) \subseteq null(A^T)$ $col(B^T) \perp col(A)$
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1answer
10 views

Finding $[T]_E$ for a basis that is composed of matrices

The linear transformation from $M_2 \rightarrow M_2 $ (Matrices that are 2x2) $T\left[\begin{pmatrix} a & b \\ 0 & d \\ \end{pmatrix}\right]=\begin{bmatrix} 1 ...
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0answers
17 views

Proving a subgroup is a basis for a space

Question: V (in R) is the subspace of all 2x2 Matrixes that are upper triangular. Prove that B is a basis for V. B= b1= $ \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ ...
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23 views

Change of Basis between linear Transformations

I am trying to get a better understanding in change of basis with matrices and linear transformations, therefore I am using several linear Transformations $^{i-1}A_i=\begin{bmatrix} \cos\theta_n ...
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2answers
17 views

Proving Rank of a matrix is greater than its sub matrix

How can I show that the rank of a matrix is always greater than or equal to the rank of every square matrix thereof.. I mean it is self evident to anyone who knows anything about rank of matrices but ...
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1answer
16 views

Assume $T$ is a complex operator

Assume $T$ is a complex operator such that $T^{2}=T$. Prove that $Tr(T)$ is a non-negative integer. There is a remark in my book, Suppose the characteristic polynomial $\chi_{T}(x)$ factors intro ...
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1answer
55 views

The tangent space of a vector space

I'm trying to show that there is a canonical isomorphism between a finite-dimensional vector space $V$ (regarded as a $C^\infty$ manifold) and its tangent space $T_vV, v\in V$, without using a basis, ...
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2answers
52 views

Efficiently solving many sets of linear equations without inversion or factorization

Suppose I have the normal set of linear equations $Ax = b$. If I can store and manipulate $A$ I have a variety of techniques available to me such as inversion, factorization, or an iterative method. ...
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0answers
28 views

Closest positive definite matrix to arbitrary one

Given a real symmetric nondegenerate matrix with complex eigenvalues. How can one find the closest (in terms of $L^2$ or $L^p$ norm) positive definite matrix? Conversely, given a non-definite ...
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1answer
32 views

Is there a bijection from 3-dimensional to 2-dimensional cartesian space?

Given a set $ M $ of coordinates in 3-dimensional cartesian space. Is it possible to find a bijection to 2-dimensional cartesian space? (This question arose from a rather practical problem of ...
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1answer
28 views

Prove that $e^{2\pi i/5}$ is not in the $7$-th cyclotomic field.

Let $\xi_n = e^\frac{2\pi i}{5} $. Prove that $\xi_5 \notin \Bbb{Q}(\xi_7)$ where $\Bbb{Q}(\xi_7)$ is the 7-th cyclotomic field. How would I approach this question? I'm having a difficult time coming ...
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25 views

Proving a basis and dimension

Hi I'm currently looking at proofs in linear algebra and came across this one and I'm compketely baffled Suppose $ C_{ij} $ is the $2\times3$ matrix with $1$ in the $ i,j^{th} $ entry and zero ...
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3answers
27 views

Proving a basis in linear algebra

So at the moment I'm trying to go through proofs and I came across this one: Suppose $ P_n $ is the vector space of all polynomials with degree less than or equal to n. Prove that $ \{1, x − 1, ...
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3answers
40 views

How can I solve for the eigenvalues for this 3x3 matrix?

I know the eigenvalues are $0, 0, 6$ but I just don't understand how it was solved. To be more precise, please explain the last few steps. The matrix is $$\begin{bmatrix} 1 & 2 & 3 \\ 1 & ...
1
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2answers
21 views

Partly solving an underdetermined system of equations

Assume that $A=\{a_{ij}\}$ is an $M\times N$ matrix where $M<N$ and $Ax=b$ where $x$ is the vector of unknown variables and $b$ is a known binary vector. Assume $a_{ij}$ values are also binary. ...
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1answer
19 views

Relation between $v$ and column space of matrix $A=(I+vv^T)$

Consider the identity matrix with a symmetric rank-one update, i.e., $A=I+vv^T$. Is there any relation between $v$ and the column space of $A$.
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1answer
31 views

Does there exist $P$ such that $PP^{\dagger}=\left(\begin{array}{cc} I & 0\\ 0 & -I \end{array}\right)$?

Does there exist $P$ such that $PP^{\dagger}=\left(\begin{array}{cc} I & 0\\ 0 & -I \end{array}\right)$? Here $P^{\dagger}$ is the hermitian of $P$, and $I$ means a $N\times N$ identity ...
2
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2answers
24 views

Reciprocal of a quadratic form

I am working with an expression of the form $$ \frac{x^TAx}{{x^TBx}}$$ and would like to simplify it. I understand that vectors do not have inverses, but viewing the bottom number as a 1 by 1 matrix, ...
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1answer
15 views

Suppose $C = AB$. Show $\hat{c}_j = \sum_{k} b_{kj}\hat{a}_{k}$.

Suppose $C = AB$. Show $\hat{c}_j = \sum_{k} b_{kj}\hat{a}_{k}$. $A$, $B$, and $C$ are square matrices of the same size. $\hat{c}_j$ is the $j$th column of $C$, $\hat{a}_k$ are the columns of $A$, ...
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0answers
30 views

1-How my profesor reach this solution? 2-How can I use eigenvalues to compute betas?… if there is any way

this time I quite don't undertand how the profesor avoid using matrix algebra to solve this exercise. Statement: Below you can see a scatter plot of the data with the three regression lines ...
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2answers
17 views

Adjoint of linear transformation $T: \mathbb{M_n(C)} \rightarrow \mathbb{M_n(C)}$

Let V = ${M_n(\mathbb C)}$ with inner product $\langle A, B\rangle = \text{tr}\,(B^*A)$, $A, B \in V$. Let $M \in {M_n(\mathbb C)}$, Define $T: V \rightarrow V$ by $T(A) = MA$. What is adjoint of ...
2
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1answer
15 views

determinant inequality for Hermitian matrix

$A \in \mathbb{C}^{M \times M}$ is a positive semidefinite matrix with all diagonal entries being $1$. and the vector $\mathbf{y} \in \mathbb{C}^{M}$ has entries $|y_{i}| < 1$. Prove that $$2 ...
3
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2answers
31 views

Property of group of $k \times k$ orthogonal matrices

Does the group of $k \times k$ orthogonal matrices lie on an $(k^2 - 1)$-sphere? If so, of what radius? If not, does it lie on some sort of other object?
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3answers
27 views

Eigenvalues of a 2 by 2 matrix

How do I show that if the eigenvalues of a 2x2 matrix A is 0 and 1, then $A^2=A$. I know that the if $A^2=A$, then the eigenvalues of A are 0 and 1. But I have no idea how to prove the this problem.
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0answers
31 views

Find a basis for subspace $W$

Let $ W=\{(a,b,c,d,e)\in \mathbb R^5:a=3c,2b-4d+5e=0\}$ I am looking for a basis for this subspace of $\mathbb R^5$. I do not remember the way to do this, can someone give me a hint?
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1answer
37 views

Vector space endomorphisms in $\mathbb{R}[x]$ commuting with $E:f\mapsto f+f'$

I am wondering if every vector space endomorphism in $\mathbb{R}[x]$ that commutes with $E:f\rightarrow f+f'$ is invertible. (denoting $f'$ the derivative of $f$) To begin with, $E$ is invertible ...
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0answers
27 views

For what positive integers $n$ and m are $\sin (nx)$ and $\sin$ $(mx)$ orthogonal over $0 ≤ x ≤ 2π?$

The question is: For what positive integers $n$ and m are $\sin (nx)$ and $\sin$ $(mx)$ orthogonal over $0 ≤ x ≤ 2π?$. Let $f(x) = \sin(nx) $ and $g(x)= \sin(mx)$ Using this formula ...
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2 views

Linear and Gondran-Minoux independence in (Max,+) Algebra

I'm reading Peter Butkovic's monograph Max-Linear Systems: Theory and Algorithms. In Chapter 6, linear independence and Gondran-Minoux independence are introduced. A set of vectors $\{a_1, a_2, \dots ...
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18 views

Properties of Determinants - True or False [on hold]

Can you help me answer these true or false questions for an n x n matrix A? I think that 3 and 10 are actually false Picture of the problem The determinant of a lower-triangular matrix A is the sum ...
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0answers
23 views

Divisibility of dimension by matrix equations.

Let ${\bf M}$ and ${\bf N}$ be two real $k\times k$ matrices such that ${\bf M}^2+{\bf N}^2={\bf M}{\bf N}$. Show that if $\det\left({\bf N}{\bf M}-{\bf M}{\bf N}\right)\neq 0$, then $3\mid k$.
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1answer
21 views

Related to inverse function theorem

We're using a book that has some really bad typos and I just want to make sure this exercise from it doesn't contain a serious one. This is the problem exactly as it's written in the text: Let $U ...
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2answers
30 views

Basis of a 3x3 eigenspace

I'm currently in the middle of a question where I'm given a 3x3 matrix: $$\left(\begin{array}{rrr} 3 & 0 & 0\\ -2 & 7 & 0\\ 4 & 8 & 1 \end{array}\right).$$ and have been ...
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1answer
15 views

Diagonal block matrices of a positive definite block matrix

Let $R=\begin{bmatrix} R^{11} & R^{12} & R^{13} \\ R^{21} & R^{22} & R^{23}\\ R^{31} & R^{32} & R^{33} \end{bmatrix}$ be a symmetric positive definite matrix where $R^{ii}$, ...
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0answers
25 views

If $v^T A^{-1} u = -1$, then the matrix $A + uv^T$ isn't invertible

Let $A \in M_n(\mathbb{R})$ be an invertible matrix and $u,v \in \mathbb{R^n}$. By the Sherman Morrison formula, we know that if $v^TA^{-1}u \neq -1$ then $(A + uv^T)^{-1}$ exists. I want to prove ...
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2answers
28 views

For $T:\mathbb{R}^6 \to \mathbb{R}^6$ and $T^5 \neq 0, \; T^6 = 0,$ prove there exists no $S$ such that $S^2 = T.$

Let $T:V \to V$ be a linear operator such that $T^5 \neq 0,$ but $T^6 = 0.$ Suppose $V$ is isomorphic to $\mathbb{R}^6.$ Prove that there does not exist an $S:V \to V$ such that $S^2 = T.$ Does the ...
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1answer
26 views

How do I determine $\min \left \| \vec{v}-\vec{u} \right \|_2$ for $\vec{u}\in U$?

Let $U=\lambda ((1, 0, 1, 0)^T,(1, 1, 0, 1)^T,(1, -1, 1; 0)^T$ is a subspace of $\mathbb{R}^4$. Determine for $\vec{v} = (1, 1, 1, 1)^T$ the vector $\vec{u}\in U$ minimal with $\left \| ...
0
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1answer
18 views

Does Linear Transformation transforms orthonormal bases of symmetric matrix into orthogonal vectors?

How would you show it? I went all over the book: looked at Linear Transformation definition again, looked at orthogonality of bases for symmetric matrices. I got to the point: Since L is linear ...
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1answer
62 views

prove that $TS = ST$

Let $V$ be a finite-dimensional vector space over $F$ with $\dim(V) = n$. Let $ T:V \rightarrow V$ be a linear transformation. Assuming that $T$ has $n$ different eigenvalues. prove that :$$ TS = ST ...
1
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1answer
14 views

Finding the inverse of a map given in vector form.

The question asks me to find the inverse map $ \mathbf\Phi^{-1} $, of: $$ \mathbf{\Phi}(\mathbf{x})= \mathbf{n\lor(x \lor n)} + \alpha\mathbf{(n \cdotp x)n} $$ for $\alpha$ such that the inverse ...