Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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The intersection between two n-spheres is a (n-1)-spheres

It is true that the intersection between two $n$-sphere is a $(n-1)$-sphere if is not empty or a single point? I have tried to prove it but my only idea is to work with equations and apparently is not ...
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5 views

Prove a certain matrix is positive semidefinte.

Consider a stochastic matrix $P$, i.e. real, non-negative, square, rows sum to one. Consider $\Xi$ to be a diagonal matrix with a principal left eigenvector of $P$ on the main diagonal and zeros ...
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1answer
16 views

Show matrix is element in eigenspace

Let $A$ be an $n\times n$ matrix such that $A^2=A$. a) Let $E_{1}(A)=\{x \in \mathbb{R^n} | Ax=x \}$: let $E_{0}(A)=\{{ x \in \mathbb{R^n} | Ax=0\}}$. Let $x$ be any vector in $\mathbb{R^n}$. Show ...
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4answers
77 views

Rank of a matrix $A^2$ without calculating the square

I have a matrix $A=\begin{bmatrix} 2 & 0 & 4\\ 1 & -1 & 3\\ 2 & 1 & 3 \end{bmatrix} $ with rank 2. How do I prove that the matrix $A^2$ has also rank 2 without actually ...
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0answers
4 views

Find coordinates of point that satisfy given conditions

I have A(1,2,3) , B(-1,0,1), C(1,-1,1) which are points in $\mathbb{R}^3$. I'm trying to find another point H such that AH${\parallel}$AC and BH${\perp}$AC. I set H = ($h_1$, $h_2$, $h_3$), and took ...
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1answer
15 views

Spectral decomposition - generalized eigenspaces

Suppose we have a real $n\times n$ matrix $A$ with spectrum $\sigma(A)=\{\lambda_1,\lambda_2\}$ (with $\lambda_1, \lambda_2$ discrete). Also, we have $alg\,mult\, \lambda_i\neq geo\, mult \,\lambda_i ...
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2answers
18 views

How to prove matrix geometric convergence to any matrix?

Suppose I have two vectors $x$ and $v$, and we want to calculate the following expression: $$(I-x\cdot v^{T})^{-1}$$ My professor affirmed that we could treat this as a "geometric progression" ...
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10 views

Eigendecomposition Parameterization of Real Matrix

Given a set of distinct non-real eigenvalues $\lambda_1, \dots, \lambda_N$, so that $\lambda_{2n} = \overline{\lambda_{2n+1}}$. Accordingly given a set of non-real orthonormal eigenvectors $v_1, ...
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0answers
31 views

Orthogonal basis of $R^n$

I have to prove that if orthogonal basis of $\mathbb R^n$ containing only vectors which coordinates are $1$ or $-1$ exisists then $n \leqslant 2$ or $n$ is divisible by 4. It's obvious that n have to ...
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1answer
16 views

Help solving system of linear equations.

In the process of running through an algorithm, I have derived the following systems of equations: i) $1/3 + 1/3x_1 + 1/3 x_6 = x_5$ ii) $1/2 + 1/4 x_6 = x_1$ iii) $1/2 + 1/2 x_5 = x_6$ I've tried ...
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2answers
47 views

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$ I'm pretty sure it's not necessarily true, but can't think of a counter example. Can you help me think of ...
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1answer
20 views

Which of the following are true?

I need to find which of the following are true? $\exists A\in M_{2\times 5}(\mathbb{R})\ni\dim$ of null space of $A$ is $2$ My ans: False as $\dim Null(A)+\dim Im(A)=5\Rightarrow\dim ...
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0answers
13 views

Verify that the set $\{ (b-a)^{-\frac{1}{2}}e^\frac{2\pi ixj}{b-a} \}$ is an orthonormal basis of $L^2(a,b)$.

Let $H=L^2(a,b)$ with $a<b$. Verify that the set $\{ (b-a)^{-\frac{1}{2}}e^\frac{2\pi ixj}{b-a} \}$ is an orthonormal basis of $L^2(a,b)$. Verify also that $$\{(b_1-a_1)^{-\frac{1}{2}} \cdot \dots ...
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1answer
33 views

understanding the matrix transpose

Can you explain me, please, what does it mean the transpose of a matrix ? I know the definition in the context of matrix theory and its generalization to adjoint operators (transpose of a linear ...
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4answers
18 views

Orthogonal nonzero vectors and linear independence. [on hold]

Show that if $\mathbf{u}$ and $\mathbf{v}$ are orthogonal nonzero vectors, then they are linearly independent.
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3answers
43 views

Question on matrix exponential

Let $A$ be a real matrix with real eigenvalues $\lambda_k$ and complex eigenvalues $\alpha_ k \pm i\omega_ k$ , all of which are simple. I'm trying to show that every element of the matrix $e^ {tA}$ ...
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4answers
80 views

How do I show that the derivative of the path $\det(I + tA)$ at $t = 0$ is the trace of $A$?

Here, I'm taking $A$ to be a linear operator on $\mathbb R^n$ for $n>1$. Can you please tell me how to solve such a problem?
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1answer
29 views

Show that $W$ is a subspace of $\mathbb{R}^n$ [on hold]

Suppose $\mathbf{z}$ is a vector in $\mathbb{R}^n$ and $W = \{\mathbf{u}\in\mathbb{R}^n:\mathbf{u}\cdot\mathbf{z}=0\}$. Show that $W$ is a subspace of $\mathbb{R}^n$.
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0answers
8 views

One-to-one correspondence between mean value and parameters

I am currently taking a course in statistics, and in this course we are considering linear models $\mu = X\beta$ where $\mu \in L$ and $L = col(X)$ is a linear subspace of $\mathbb{R}^n$, $X$ is the ...
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1answer
25 views

Reduce matrix to Smith Normal form.

I've been given the finitely generated abelian group: $$\langle x_1, x_2 \mid 6x_1-6x_2, -6x_1-12x_2, 4x_1-8x_2\rangle$$ and written the corresponding matrix: $$A=\begin{pmatrix} 6 & -6 \\ -6 ...
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1answer
21 views

Maple - Substitute into an expression involving derivative

I'm working with some matrices including derivatives like d/dt(x(t)). I need to replace this whole expression {d/dt(x(t))} with something like xdot. I have tried to use "subs", but it seems to refuse ...
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9 views

Analytical solution to the first PCA direction

It is known that the first PCA direction for a dataset of $n$ points is the unit vector with max variance after projecting the points onto this vector. I wonder whether there are some analytical ...
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2answers
36 views

Let $A_{j,k} = \langle x_j, x_k\rangle$. Show $A$ is invertible if and only if $x_1, \ldots, x_n$ are linearly independent.

Let $V$ be a vector space over $\mathbb C$ with inner product $\langle, \rangle$ and let $x_1, \ldots, x_n$ be vectors in $V$. Consider the $n \times n$-matrix $A$ with entries $A_{j,k} = \langle ...
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2answers
18 views

Relation between the determinant of a linear mapping and norm of a multiplied vector

Let $S,T:\mathbb{R}^n \to \mathbb{R}^n$ be linear mappings with $\|Sv\|\le \|Tv\|$ for all $v\in\mathbb{R}^n$. Is it generally true that $|\det(S)|\le |\det(T)|$?
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1answer
8 views

P - bounded polyhedron, L - linear map. Show that L(P) is a bounded polyhedron

Let $P = \{x\in \mathbb{R}^n \ | \ Ax\leq b\}$ be a bounded polyhedron. Let $L:\mathbb{R}^n \rightarrow \mathbb{R}^k$ be a linear map. Show that $L(P):=\{L(x)\ | \ x\in P\}$ is a bounded polyhedron. ...
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0answers
31 views

$(v_1,v_2,v_3)$ is positively oriented iff $\det(A) > 0$

Suppose $(v_1,v_2,v_3)$ are three linearly independent vectors in $\mathbb{R}^3$. Suppose $A = \left( \begin{array}{ccc} | & | & | \\ v_1 & v_2 & v_3 \\ | & | & | ...
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How to show that the rotation around $m_0$ about an angle $φ$ is an affine map? [on hold]

Let $E^2$ be the two-dimensional Euclidean space (Euclidean plane), and let $m_0 = (1, 0)$. Show that the rotation around $m_0$ about an angle $φ$ is an affine map. Give a formula for this map with ...
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0answers
18 views

How are $EA$ and $AE^T$ related, where $E$ is an elementary matrix?

If $E_{n\times n}$ is an elementary matrix, and $A_{n\times n}$ is any matrix, how are $EA$ and $AE^T$ related? I understand that $EA$ is a row operation on $A$ and $AE^T$ is the same operation on ...
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1answer
27 views

Proof of Vector Space Axioms [on hold]

Where can I find detailed proof of vector space axioms? Any reference to a book, website or video lecture.
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0answers
8 views

4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf z} \in C^{n×1}$ is a CSCG random vector denoted with $C (μ,Σ)$ where $μ$ and $Σ$ are mean and contrivance matrix, respectively, and defined as $μ=E({\bf z})$, $Σ=E({\bf z}{\bf ...
3
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0answers
58 views

If $n$ is even, every skew-symmetric $n\times n$ matrix $A$ can be factored as $A=SBS^T$

If $n$ is even, every skew-symmetric $n\times n$ matrix $A$ can be factored as $A=SBS^T$ where $S$ is a invertible matrix and $B$ has the form $B = \left( \begin{array}{ccc} 0 & a_1 & 0 & ...
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0answers
29 views

Proving that elementary row operations are preserved after multiplication

If $E$ is an elementary $n \times n$-matrix, show that if $A$ is any $n\times n$-matrix, then $EA$ is a matrix obtained by carrying out a single elementary row operation on $A$, and that $AE$ is a ...
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1answer
11 views

If we have a singular matrix, what can we say about it's adjoint [on hold]

We need to prove that for singular matrix determinant of its adjoint is also 0.
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1answer
28 views

Proof that an involutory matrix has eigenvalues 1,-1

I'm trying to prove that an involutory matrix (a matrix where $A=A^{-1}$) has only eigenvalues $\pm 1$. I've been able to prove that $det(A) = \pm 1$, but that only shows that the product of the ...
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0answers
19 views

Do prove in vector space (about span and subspace) [on hold]

How to solve part (a) and (b) How to solve part (a) and (b) How to solve part (a) and (b) How to solve part (a) and (b) How to solve part (a) and (b) How to solve part (a) and (b) How to solve part ...
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0answers
21 views

Formula for Area of parallelogram induced by linear operator

I'm given that the linear operator $L: \mathbb R^2\to\mathbb R^2$ is invertible. The set (u,v) is a linearly independent set in $\mathbb R^2$. I must find a formula for the area of the parallelogram ...
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2answers
15 views

Find a Cartesian Equation for the Plane Satisfying Those Properties

Find the Cartesian equation of the plan parallel to j and passes through the intersection of the planes described by the equations x + 2y + 3z = 4, and 2x + y + z = 2. I was able to get the ...
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1answer
27 views

If $A$ is skew-symmetric, then a fixed row/column operation produces a new skew-symmetric matrix

Suppose $A$ is a skew-symmetric matrix. Fix an elementary row operation. If we carry out this row operation on $A$, and then carry out the corresponding column operation on the resulting matrix, do we ...
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1answer
12 views

Possible values of nullity in 4x2 matrix

Let $A$ be a 4 by 2 matrix. Explain why the rows of $A$ must be linearly dependent. What are the possible values of nullity(A)? I understand the first part. I do not understand the second part. The ...
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Given a singular covariance matrix of a random vector $X=[X1,X2,X3,X4]$, how do I partition $X$ into 2 parts to get a non-singular covariance matrix?

Suppose I have a covariance matrix of a random vector $$X=\begin{bmatrix}X1\\ X2\\ X3\\ X4 \end{bmatrix}$$ $$C_X=\begin{bmatrix}a&b&c&d\\ e&f&g&h\\ i&j&k&l\\ ...
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1answer
35 views

Show that $EA$ is obtained from an elementary row operation on $A$

Suppose $E$ is an elementary $n \times n$-matrix. Prove that if $A$ is any $n\times n$-matrix and $E$ is any elementary matrix, then $EA$ is a matrix obtained by carrying out a single elementary row ...
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2answers
38 views

To find Basis and kernel of matrix A

Given a matrix $A:$ \begin{pmatrix} 1 & 2 & 5 \\ 3 & 5 & 13 \\ -2 & -1 & -4 \end{pmatrix} My textbook has reduced it to RREF to find kernel and dimension of it. To find the ...
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1answer
33 views

An inequality about Hermitian matrices

Say one knows the following statement, That for any Hermitian matrix $H$ with eigenvalues $\lambda_1 \geq \lambda_2 ..\geq \lambda_n$ one has, that in any basis, for any positive integers $1 \leq i_1 ...
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1answer
47 views

Decompose $\omega:= e_0\wedge(e_1\wedge e_2 + e_3\wedge e_4)$

$(e_1\wedge e_2 + e_3\wedge e_4)$ is well-known to be of rank 2 (can't be decomposed). On the other hand, $\omega \wedge \omega = e_1\wedge e_1 \wedge ... = 0$. According to the wikipedia article ...
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1answer
42 views

Show that the set $\{v_n\} \subset l^2$, $v_j$ is orthonormal.

Show that the set $\{v_n\} \subset l^2$, where $v_n=\frac{1}{\sqrt{2}}(e_n-e_{n+1})$ if $n$ is odd and $v_n=\frac{1}{\sqrt{2}}(e_n+e_{n-1})$ if $n$ even is orthonormal. Is it a Schauder basis of ...
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1answer
22 views

Tricky change-of-basis transformation problem

I have absolutely no idea what to do here because of the $\sin(x).$ Let $V = \text{Span}\left\{x, x^3, \sin(x) \right\}$, and consider the basis for $V$ given by $\beta = \left\{x-2x^3, x^3+\sin(x), ...
0
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1answer
15 views

Necessary and sufficient conditions for an adjoint of a linear map to be the map's inverse

Let $V$ be a finite dimensional inner product space, $ \phi :V \rightarrow V$ a linear operator and $\phi^*:V \rightarrow V $ its adjoint. I wish to show: $\phi^*$ is an inverse to $\phi$ if and ...
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1answer
40 views

Can -3 and 2 be eigenvalues of the following matrix?

Can $-3$ and $2$ be eigenvalues of and nxn matrix B such that $A = B^{2}+B-6I$ and A's determinant is $0$? So this is what I concluded: At first glance, it can be seen that the matrix $A$ can be ...
0
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0answers
16 views

How to show that $F(m_0+v)=m_0+A(v), v∈V$ defines an affine map of $(M,V)$? [on hold]

Let$(M,V)$ be an affine space, and let $m_0 ∈ M, A ∈ L(V)$. I need to show that the equation $$F(m_0+v)=m_0+A(v), v∈V$$ defines an affine map of $(M,V)$ with linear part $LF = A$ that has $m_0$ as a ...
0
votes
2answers
28 views

Calculate the tensor product of two vectors

Let $\{e_1, e_2\}$ and $\{f_1, f_2, f_3\}$ the canonical ordered bases of $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively. Find the coordinates of $x \otimes y$ with respect to the basis ...