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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Does basis of eigenspace mean the same as eigenvectors?

If you have a 3x3 matrix, 2 eigenvalues (one with multiplicity 2) and now 2 eigenvectors, how do you find the basis for each eigenspace?
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Orthogonal Matrices and Similarity Transforms

Sorry I can't be more specific with the title. I really don't know what to call this and about 2 hours of Googling has yielded no results. All we are given: $U$ is $n\times n$ and orthogonal $Ax = ...
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Can the number of sign changes in a sequence of determinants tell us how many negative eigenvalues a symmetric matrix has?

From notes, I've gathered that given a symmetric matrix, the number of sign changes in its characteristic polynomial is equal to the number of positive eigenvalues of $A$. Proof: Let $p(x)$ be a ...
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1answer
14 views

Uniqueness of Thin QR Factorization.

Let $A \in \mathbb C^{m x n}$, have linearly independent columns. Show: If $A=QR$, where $Q \in \mathbb C^{m x n}$ satisfies $Q^*Q=I_n$ and $R$ is upper triangular with positive diagonal elements, ...
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25 views

Simultaneous function with three variables using subsititution method

Use any substitution method and solve the following equations: $$2x+5y+7z=86$$ $$3x+y+5z=60$$ $$x+4y+3z=54 $$ I used $x+4y+3z=54$ to make $x$ the subject $x=54-4y-3z$.
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1answer
11 views

Covariance matrix of Y when we have the covariance matrix of X

If the random vector $\mathbf{X}$ is transformed according to \begin{align*} Y_1 &= X_1\\ Y_2 &= X_1 + X_2 \end{align*} and has a covariance matrix $$ \mathbf{C}_X = ...
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2answers
27 views

Does the line $(2,1,1)+t(-3,1,5)$ live within the plane $31x+3y+18z=62$?

I have a doubt with this exercise: Have the plane $$31x+3y+18z=62$$ What is the distance between this plane and some line $(x,y,z) = (2,1,1) + t(-3,1,5)$ for some $t\in\mathbb{R}$? The ...
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29 views

Differences of grade between this three books

I have a simple question, I noticed these three books for my study, but I didn't understand the grade of these books because the names of the paragraphs are similar . 1) ...
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1answer
18 views

Rank of a special matrix

Say a $5\times 5$ matrix $$A = \left[ \begin{array}{ccc} 1&2&3&4&5\\ 6&7&8&9&10\\ 11&12&13&14&15\\ 16&17&18&19&20\\ ...
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2answers
24 views

How to show that $\| QA\|_2=\| A \|_2$ where $Q$ is unitary (for a matrix A)

I want to show that for a unitary matrix $Q$ and a matrix $A$ that $$ \|QA\|_2=\|A\|_2$$ I start with the definition of matrix induced norms: $$\| QA \|_2 = \sup_{x \neq ...
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How to show that the surjectivity of a linear map $f: R^ n\to R^n$ implies the injectivity and vise versa?

How to show that the surjectivity of a linear map $f: R^ n\to R^n$ implies the injectivity and vise versa?
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2answers
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Semisimple modules and the radical

I don't need a proof, but can someone tell me whether it is true that for all $A$-modules $V$ we have that $V/\text{rad}V $ is semisimple, where we define $\text{rad} V$ as the intersection of all ...
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20 views

Number of Jordan canonical form of a matrix

Let, $A\in M(3,C)$. Assume that the characteristic & minimal polynomial of $A$ are known. Then what is the number of possible Jordan form of $A$ and how? What changes if we replace $C$ by $R$ or ...
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0answers
7 views

Eigensystem of a real symmetric Toeplitz matrix of large order

My question is related to this one. I am looking for the eigenvalues and eigenvectors of a square, symmetric, real Toeplitz matrix of order N where N is large. There are some references in the above ...
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1answer
29 views

Are the following quotient spaces finite dimensional?

If we take $\mathbb{F}[x]$ to be the set of all polynomials over the field $\mathbb{F}$, $E$ to be the subset of all such even polynomials, $N$ to be the set of these polynomials that have degree less ...
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0answers
12 views

Generalize discrete Lyapunov equation for n-th order linear dynamics system

My specific application is analysis of dynamic textures using linear dynamics systems of the form $$ I(t) = Cz(t) + w(t) \\ z(t + 1) = Az(t) + Bv(t), $$ where $I(t)$ is the original signal, $z(t)$ ...
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1answer
26 views

A question on a nonnegative quadratic form

Denote $x,y,z$ as variables, and $a,b,c$ as coefficients. Suppose $a\leq b\leq 0\leq c$ and $a+b+c=0$. Could anyone help me prove whether the following quadratic form positive semi-definite? ...
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2answers
15 views

Finding the co-ordinate vector

I can find the co-ordinate vectors for all $x$ in $R^n$ but I can't wrap my head around the ones for $x$ in $P_n$. Here is a question: Let $V$ be the space $P_3$ of all polynomials of degree at ...
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1answer
13 views

What is the relationship between parallelogram law and polarisation identity?

According to wikipedia article on polarisation identity, in a normed space $(V, || . ||$), if the parallelogram law holds, then there is an inner product on V such that $||x||^2 = \langle x, x\rangle$ ...
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1answer
15 views

Speed of two points on a circle

Problem Two points $A$ and $B$ are moving on a circle at constant speeds $v_A$ and $v_B$. We assume that they start from the same position and that they instantly accelerate to their final speed. ...
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7 views

Proof: F is isometric if and only if its matrix is orthogonal/unitary

I'd like to show that $F \in End(V)$ isometric $<=> M_{\beta \beta} (F)$ orthogonal/unitary But it seems as if I still have some trouble doing that ;/ "=>" $<v_i, v_j> = ...
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In a vector space $V$, If $v^tAw = v^tBw$ for all $v,w \in V$, does it imply $A=B$?

OK. So, basically this question came up when I was trying to solve a homework question about how the matrix representation of a bi-linear form on $V$ changes if we change the basis on $V$. Let's say ...
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273 views

A method of finding the eigenvector that I don't fully understand

Let $$A=\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & t \\ \end{pmatrix}$$ Which has a known eigenvalue : $\lambda$ Find the corresponding eigenvector Over the ...
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1answer
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Help me understand Vector Spaces (proving linear spaces)

Please help me understand each part clearly. Please don't give general answers, it's easier for me to understand concepts by doing specific questions and learning about them. (i) The reason ...
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1answer
18 views

By given equation, finding orthogonal projection

Find the orthogonal projection of line with direction vector $u = ( 1 , 2 , 0 )$ onto the plane described by equation $-3x - 2y + 2z = -2$ i have tried to search ...
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3answers
27 views

How to determine volume of parallelepiped by 4 points

Let points $(0,0,0), (1,2,x), (-2,1,0)$ and $(1,1,3)$ be at four corners of a parallelpiped. Determine the volume of the parallelepiped by using the determinant in terms of $x$. For what value of $x$ ...
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3answers
47 views

What does inner product actually mean?

What does inner product actually mean? So far most of the cases that I encounter seems to suggest that dot product is the only useful inner product. I mean most of the things that we discuss about ...
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18 views

Which curve we will get under $\mathcal{A} \in M_2(\Bbb{R})$ from a unit circle

If I have a circle: $x^2+y^2=1$, It's parametric equation is : $$\begin{cases} x = \cos\theta \\ y = \sin\theta \end{cases}$$ under some transform: $A=\begin{pmatrix} a & b\\ c & d ...
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22 views

Speeding up solving a linear system

I need to speed up calculating the following linear system: $$ (A^TA +\rho I + \nu \sum_{k=1}^l (q_{k,1}q_{k,2}^T+q_{k,2}q_{k,1}^T))x=b, $$ where $A\in\mathbf{R}^{m\times n}$, $\rho,\nu$ in ...
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2answers
51 views

determine signature of matrix

what is the signature of this matrix: $\begin{pmatrix} -3&0&-1 \\0&-3&0 \\ -1&0&-1 \end{pmatrix}$ ? I tried calculating them without eigenvalues; this should be done via ...
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1answer
19 views

Why isn't the orthogonal vector to a direction vector of a plane not necessarily perpendicular to such plane?

I had made a question, and the problem with my exercise was that I was trying to calculate a vector perpendicular to some plane in $\mathbb{R}^3$: given one line $L$ inside the plane, I grabbed the ...
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3answers
54 views

Does there exists a vector v such that $Av\neq 0$ but $A^{2}v=0$

Let A be a $4\times4$ matrix over C such that $\operatorname{rank}A=2$ and $A^{3}=A^{2}\neq0$. Suppose that A is not diagonalizable. My question is , "Does there exists a vector $v$ such that ...
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0answers
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How to create an equation from this problem?

A full cola bottle is $2. 2 caps can be exchanged with 1 full cola bottle. 4 empty bottles can be exchanged with 1 full cola bottle. If you have $20, how many full coke bottles you will totally ...
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0answers
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Matrix Decomposition: Difference between Cholesky Decomposition, Eigendecomposition and Jordan Normal Form Decomposition

I recently created a related topic about the square root matrix, in case you'd like to refer to that one. Here's what we want: Consider the matrix $\Omega=E(\mathbf{u}^{\top}\mathbf{u})$, where ...
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1answer
18 views

Inner product and unit vector

$u_1 = (1, -1)'$ and $u_2 = (1, 1)'$ are two vector of $R^2$. Endow $R^2$ with an inner product such that $u_1 = 1$ and $u_2 = 1$. Well, honestly, I don't completely understand what the problem asks. ...
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2answers
66 views

Show matrix is positive [semi]definite

I want to show $H = (1-\rho)\mathbf{I} + \rho\mathbf{1}\mathbf{1}^\intercal$ is positive [semi]definite where $$\dfrac{-1}{n-1} < \rho < 1$$ where $\dim H = n\times n$ So far I have, for any ...
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0answers
34 views

Prove that there is a subset with an invertible sum

The question is as follows: For some $m>n$, let $A_1,\cdots,A_m$ be $n\times n$ matrices, satisfying $$ A_1+\cdots+A_m=I_n $$ where $I_n$ is the $n\times n$ identity matrix. Show that ...
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1answer
64 views

any symmetric matrix is ​​invertible?

is a simply theoretical question, but any symmetric matrix is ​​invertible? i'm trying to prove this question but I don't know what I need to do. I apologize for the simple question but is a doubt ...
2
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3answers
49 views

The normal equation of the plane that contains the line $(1,1,1) + t(-2,0,3)$

Determine the equation of the plane that contains the point $(4,2,-1)$ and also the line $L: (1,1,1) + t(-2,0,3)$ for $t\in\mathbb{R}$. The direction vector $(-2,0,3)$ of the line is also a ...
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2answers
21 views

Find one set of solutions for the following system:

Find one set of solutions for the following system: \begin{cases} 1+a^2+d^2=3+b^2+e^2=3+c^2+f^2 \\ 1+ab+de=0 \\ ac+df=0 \\ bc+ef=0 \\ \end{cases}
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1answer
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How do I find a constant for a polynomial so its roots are reflective around a linear function?

How can I find all complex numbers $w$ so that the roots of the following polynomial are reflected around a linear function $f(x)$ $$p(q) = q^2-4q+w = 0$$ If I want to find all the complex numbers ...
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1answer
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If $A_{n\times n}$ and $B_{n\times n}$ are both nonsingular real matrices, where $n$ is odd, show that $AB + BA \neq0$.

I have been puzzling over this for a while now. I tried to find something in the properties of nonsingular matrices as well as the properties of determinants that might relate, but so far I've found ...
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1answer
28 views

The condition that $[LM:K]=[L:K][M:K]$ holds in the field.

Let $L$ and $M$ be intermediate fields of the extension $K\subset F$, of finite dimension over $K$. Assume that $L\cap M=K$ and $[L:K]$ or $[M:K]$ is 2, then $[LM:K]=[L:K][M:K]$ holds. How can I use ...
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0answers
32 views

Solving matrix equation of the form $(AX)^2+(BY)^2=D$

Is there any method that can solve the matrix equation of the form $(AX)^2+(BY)^2=D$? $A$ and $B$ are matrices, $X$, $Y$ and $D$ are column vectors. (Solve for $X$ and $Y$) I originally have two ...
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1answer
28 views

About kernel space

Both the square and symmetric matrices $A$ and $B$ are positive semidefinite. Moreover, $A-B$ is positive semidefinite and $\text{rank}(A)=\text{rank}(B)$. Based on these conditions, can we have ...
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11 views

Show that a constructed matrix is a unitary matrix

Given that $\{|\psi_i\rangle\}$ and $\{|\phi_i\rangle\}$ are sets of orthonormal eigenvectors, show that a matrix $$ M = \sum_i{|\psi_i\rangle \langle \phi_i|}$$ is a unitary matrix. I have tried the ...
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1answer
32 views

Determining if one is a subspace

Define the following linear mappings: $$L:R^n→R^m$$ $$M:R^m → R^P$$ Prove that Range $(M◦L)$ is a subspace of Range $(M)$. What I have so far (not sure if correct): Range $(M◦L)=R^p$ and Range ...
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2answers
47 views

How do i solve this equation? [on hold]

I need help with one of the equations that I'm going to have on my test: $$5-2x-\frac{5-3x}{2}=1$$
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0answers
21 views

How to find the basis of a matrix by using Gauss-Elliminaton?

I confuse that, This is my calculating process, Where i do the mistake in this process? I hope to understand this error.
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3answers
40 views

Linear system $AX=0$ has a nonzero solution

Which parameters $a,b,c,d$ satisfy the matrix $$A=\begin{bmatrix}-1 & 1&1&1 \\1 & -1&1&1\\1 &1 &-1 &1\\a&b&c&d \end{bmatrix}$$ sucht that linear system ...