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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3
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3answers
56 views

Why is cross product not commutative?

Why, conceptually, is the cross product not commutative? Obviously I could simply take a look at the formula for computing cross product from vector components to prove this, but I'm interested in why ...
5
votes
4answers
947 views

The contributions of James Sylvester to linear algebra.

The claim is James Sylvester and Arthur Cayley are the fathers of Linear Algebra. I can find the various parts that Cayley contributed to Linear Algebra, but there is not much on the contributions ...
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0answers
13 views

finding equality with subspaces direct sum

assume that $U_1 \cap U = \{0\}$ and $U_2 \cap U = \{0\}$ $U_1 \oplus U = U_2 \oplus U$? I thought that it's correct because I could find a counterexample.
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3answers
39 views

Linear Dependent Span

$\{x \cos x, x, \cos x \}$ is a subspace of $V$. I need to find if it's a linear dependent or linear independent. So I thought that its dependent since $x \cos x$ is multiplication of $x$ and $\cos ...
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0answers
23 views

Orthogonal Projections Composition iff Statement [duplicate]

Given m , n ⊂ V a finite dimensional inner product vector space, Prove that for Pm, Pn orthogonal projections onto m, n accordingly, PmPn is an orthogonal projection ⇔ PmPn = PnPm and in this case ...
1
vote
1answer
23 views

Norm of a complex cross product

Let $c=(c_1,c_2,c_3)$ be a complex vector. How can we see that $\|c\|^2=\|c\times \bar{c}\|$? Here the bar means component wise complex conjugation, the norm is the Hermitian norm, and the cross ...
0
votes
1answer
34 views

Similarity of a specific block matrix

Let $A$ and $B$ be square matrices of same dimension. I considered $n$-by-$n$ block matrices of the form \begin{align*} \begin{pmatrix} A & & \\ & \ddots & \\ & & A ...
1
vote
1answer
41 views

Spectrum of the matrix $A=(a_{ij})$ where $a_{ij}=i+j$

What is the spectrum of the matrix $A=(a_{ij})_{n\times n}$ where $a_{ij}=i+j$ for any $n$. Also, what are the eigenvectors corresponding to their eigenvalues? Progress. This matrix is definitely ...
-1
votes
1answer
15 views

an inequality for the projection on the intersection of 2 subspaces

Can someone give an inequality, bounding the distance of a point from its projection on the intersection of two subspaces by a function of both the distances of the point from the insividual ...
1
vote
1answer
32 views

What are the facts used in each step of this proof?

What are the facts used in each step of this proof ? Suppose that $A\in F^{nm}$ and $B\in F^{ml}$ $$\begin{align}rank A + rank B &= rank\begin{bmatrix}0 & A\\B & 0\\ \end{bmatrix}\\ ...
1
vote
4answers
30 views

Does an arbitrary matrix $X \in M_{n \times p}$ have a SVD?

I have proven, as below, that if $X \in M_{n \times n}$ is symmetric, then it has a SVD. $D(\lambda_i) = \text{Diag}(\lambda_i)$ is a diagonal matrix with entries $\lambda_1, \lambda_2, \dots$. ...
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vote
0answers
18 views

Hermitian, orthogonal, unit trace matrix bases.

Consider the vector space of Hermitian matrices acting on a finite dimensional vector space, equipped with the Hilbert-Schmidt norm. I'm interested in matrix bases that satisfy three properties: 1) ...
0
votes
2answers
41 views

Help understanding the range and kernel of a linear transformation

I'm having some trouble understanding the Range and Kernel of a linear transformation. The definition goes as follows: Let $T:V \longrightarrow W$ be a linear transformation. Define the sets ...
0
votes
0answers
15 views

find Jordan form

Determine the jordan form of $A = \begin{pmatrix} 1 & 2 & 3\\ 0 & 4 & 5\\ 0 & 0 & 4 \end{pmatrix} $ First, I find the characteristic polynomial. $C_A(x)=(x-1)(x-4)^2$. ...
0
votes
0answers
15 views

upper bound for the sum of trace related to product of two matrices?

Given A and B positive definite matrices The inequality $\sqrt{4tr(AB)}$ $\leq$ $tr(A+B)$ is lower bound for tr(A+B) is there another inequality for the upper bound, i.e. ?? ≥ tr(A+B)?
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vote
3answers
211 views

Help determining whether a transformation is linear or not

I have the following transformation: $T : \Bbb R^3 → \Bbb R^2 , T (x_1 , x_2 , x_3 ) = (x_1 − x_2 , 2x_2 )$ I need to determine whether it's a linear transformation or not. I understand that ...
2
votes
0answers
36 views

Second order derivation of Quadratic form

I would like to find the second order derivative of a Quadratic form. Assume we have a random complex column vector $x$ and a real constant value $C$. I am interested in computing the following: $$ ...
0
votes
1answer
39 views

Prove that if $A,B\in M_n(\mathbb{F})$ are $(n-1)$-nilpotent then they are similar.

If $A,B\in M_n(\mathbb{F})$ are $n-1$ nilpotent, prove they are similar. Can I say that, since their minimal polynomial is $X^{n-1}$ they are similar? I know that If $A,B$ are similar, they have ...
0
votes
2answers
35 views

Find the standard matrix for the transformation [on hold]

Find the standard matrix for the transformation that projects $$ \begin{bmatrix} 1\\ 2 \\ \end{bmatrix} \text{ to } \begin{bmatrix}2\\ 4 \\ \end{bmatrix} $$ and $$ \begin{bmatrix} 1\\ 1 \\ ...
7
votes
2answers
92 views

$A^2=A^*A$.Why $A$ is Hermitian matrix?

Let $A$ be $n \times n$ matrix and $A^2=A^*A$. Why is $A$ a Hermitian matrix?
3
votes
1answer
69 views

Matrix and field extension

It is given that $F\subset K$ are fields. $A$ is a matrix of size $n\times n$ over $K$. I need to prove that there exist $c_1,\ldots,c_k\in K$, linearly independent over $F$, and matrices ...
0
votes
3answers
37 views

How to find the basis of the following vector spaces?

I'm trying, in vain, to find the basis of the following vector spaces: (a) $W = \{x = (x_1 , x_2 , x_3 ) ∈ \Bbb R^3 : x_1 − 2x_2 + x_3 = 0, 2x_1 − 3x_2 + x_3 = 0\}$ (b) $W = \{x = (x_1 , x_2 ...
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votes
0answers
20 views

Relation between Ill-posed problem and eigenvectors?

This question is related to the question below: Is there a relation between Ill-posed problems and Eigenvectors. In the answer of the above question, it was shown that the ill-posed problem can be ...
2
votes
5answers
118 views

Can systems of 3 linear equations with 3 unknowns have more than one solution?

In each part,determine whether the given vector is a solution of the linear system \begin{align} 2x-4y-z&=1\\ x-3y+z&=1\\ 3x-5y-3z&=1 \end{align} (a) $(3,1,1)$ (b) $(3,-1,1)$ (c) ...
2
votes
1answer
40 views

Why does ${\lambda _{\max }}(A) = \max \{ \frac{1}{{{x^*}x}}:{x^*}Ax = 1\} $?

Let $A \in {M_n}$ be hermitian and suppose that at least one eigenvalue of $A$ is positive ($\lambda $ is eigenvalue of $A$). Why does ${\lambda _{\max }}(A) = \max \{ \frac{1}{{{x^*}x}}:{x^*}Ax = 1\} ...
2
votes
2answers
216 views

Show that 1 and -1 are the only eigenvalues of this linear transformation

Define $T: M_{n\times n}\to M_{n\times n}$ by $T(A):= A^t$. Note that $T$ is a linear transformation. Show that $1$ and $-1$ are the only eigenvalues of $T$. Let $\lambda$ denote an eigenvalue ...
2
votes
3answers
3k views

Does common eigenvectors between two matrices A,B implies some property for the vectors?

If there are two matrixes that they have common eigenvectors for some eigenvalues that implies that those two matrixes are identical? What can we say for those two matrices?
6
votes
2answers
78 views

linear map $f:V \rightarrow V$, which is injective but not surjective

I am trying to find a linear map $f:V \rightarrow V$, which is injective but not surjective. I always thought that if the dimension of the domain and codomain are equal and the map is injective it ...
2
votes
0answers
35 views

Is there an easier way to show that a kite has perpendicular diagonals using scalar products?

I want to use scalar products to prove that a kite has perpendicular diagonals. My attempt : Let $a,b,c,d$ vectors with $a+b+c+d=0$ and $a^2=d^2$ and $b^2=c^2$ Then, we get ...
0
votes
2answers
28 views

Prove every isometry on an odd- dimensional real product space has 1 or -1 as an eigenvalue.

This is a question from Axler. I was hoping for some help. It seems easy to understand, but I don't know where to go about on proving this.
0
votes
0answers
20 views

Linear Algebra L2 minimization

Im really confused about how to solve this question or even what its asking. Any help would be much appreciated! Let A be an m x n real matrix ($m \gt n$). Let x* be the minimizer of $||Ax - b||^2 + ...
0
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0answers
17 views

Circulant matrix - Eigen decomposition in matlab [on hold]

$A = FD{F^{H}}$ , where $A$ is a circulant matrix, $F$ is normalized FFT matrix (unitary) and $F^{H}$ is the conjugate transpose (Hermitian) of the $F$ matrix. $D$ is the diagonal matrix which ...
0
votes
0answers
13 views

congruent matrices

If the rank of a matrix A is 1, the matrix is row equivalent to one with only one row different to 0. But not necessary the matrix A is congruent to one wuth only one row different to 0/ How to find a ...
1
vote
2answers
327 views

Barycentric coordinates of a triangle

I have to do what described in the picture below. Consider the planar triangle $[p_1,p_2,p_3]$ with vertices $p_1=\begin{pmatrix}-2\\-1\end{pmatrix}$, ...
0
votes
2answers
172 views

Does an overdetermined system always have no solutions? [on hold]

What is the problem with over-determined systems in linear algebra? Do they always have no solution? Is there a proof of that?
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2answers
40 views

Straight line equation is linear or not?

I read somewhere that for the linearity the equation should pass through the origin in this regard the equation of straight line y=mx+c is linear or not?
1
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1answer
17 views

derivative of gradient involving inverse of matrices

I need to take three partial derivatives of this squared mahanalobis distance with respect to these three matrices: $Q, A,$ and $S$ $$(x+Ab)^T(A^TQA+S)^{-1}(x + Ab)$$ $x$ and $b$ are vectors of ...
0
votes
0answers
9 views

Exponential Demand Periodic Review

I have exponentially distributed demand data and I am trying to find a formula for an 'order up to level (OUL)' periodic review ordering policy. We are not using a re order point for this policy. ...
1
vote
1answer
750 views

Find the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$.

Find a vector equation and parametric equation of the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$. Find two points on the line that are ...
0
votes
0answers
29 views

A inquality in matrix norm [duplicate]

Let $A,I \in {M_n}$($I$ is identity matrix) and $\left| {\left\| . \right\|} \right|$ is matrix norm.Suppose $\left| {\left\| A \right\|} \right| < 1$ and $\left| {\left\| I \right\|} \right| \ge ...
0
votes
0answers
22 views

Questions about the position of a matrix and a vector

I got quite confused by position of a matrix and a vector. For example, the definition of a range space put a matrix in front of a vector, like $R(A) = \{ Ax | x∈ R^n \}$; However for a linear ...
0
votes
1answer
44 views

Is there a function over $\mathbb{Z}_p$ that is never linear?

Let $p$ be a prime. I wonder if there is a function $f$ that satisfies the following rule: Whenever $$z_1 + \dots + z_c \equiv cx \mod p$$ (where $1 < c < p$, and $z_j, x \in \mathbb{Z}_p$) ...
1
vote
1answer
24 views

How to find the span for a linear transformation?

I'm learning Linear Transformations and I understand what is a linear transformation. Now I'm trying to look at an example question and I'm not really sure how the span is found. The question goes as ...
1
vote
1answer
7k views

Magnitude of a Matrix?

Consider a vector V. The magnitude of this vector (if it describes a position in euclidean space) = distance from the origin is simply: $(V^TV)^{1/2} $ aka the square root of the dot product... ...
6
votes
0answers
56 views

Let $\mathbb{K} $ be a field of characteristic $p>0$ and $\mathbb{F} | \mathbb{K} $ a finite and separable extension.

Let $\mathbb{K}$ be a field of characteristic $p>0$ and $\mathbb{F}/ \mathbb{K}$ a finite and separable extension. Show that if $B=\{\alpha_1,\dots,\alpha_n\}$ is a basis, then ...
1
vote
4answers
45 views

Prove that $\|v \|^2= |\langle v, e_1 \rangle |^2 + \cdots + | \langle v, e_m\rangle |^2$

Suppose $(e_1,\cdots, e_m)$ is an orthonormal basis in $V$. Let $v \in V$ . Prove that $\|v\|^2= |\langle v, e_1 \rangle |^2 + \cdots + | \langle v, e_m\rangle |^2$ Let $v\in V$ and ...
0
votes
1answer
18 views

Clustering of vectors via inner product relationship

This might be an odd question, but suppose I have a lot of vectors $a_i\in\mathbb{R}^{3}$ (not necessarily unit) and for some unit vector $u\in\mathbb{R}^{3}$ I find $$ \sum_{i=1}^m ...
0
votes
1answer
33 views

column space of a matrix

If $A\in M_{m\times n}\mathbb{(R)}$, show that $\mathcal{R}(AA^t)=\mathcal{R}(A)$ and $\mathcal{R}(A^tA)=\mathcal{R}(A^t)$ where $\mathcal{R}$ denotes the column space of matrix. How can I prove it ...
0
votes
1answer
33 views

Strategies for linear systems

Consider I have the following equations. Is there a faster way for me to solve the system without going through a series of substitutions? $$-20a+13b+13c=0$$ $$10a-26b+13c=0$$ $$10a-13b-16c=0$$ ...
30
votes
5answers
19k views

How do I tell if matrices are similar?

I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not. I solved this by using a matrix called $S$: $$\left(\begin{array}{cc} a& b\\ ...