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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A question on a nonnegative quadratic form

Denote $a,b,c,x,y,z$ as variables, and $A,B,C,X,Y,Z$ as coefficients. Suppose \begin{equation*} \begin{split} &A+B+C=0,\\ &X+Y+Z=0,\\ &A\leq B\leq 0\leq C,\\ &X\leq Y\leq 0\leq Z. ...
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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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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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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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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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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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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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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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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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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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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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19 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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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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18 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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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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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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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
17 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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60 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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26 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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62 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 ...
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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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20 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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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
26 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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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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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
25 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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1answer
24 views

Zariski-open subset in $\mathbb{C}^n$ to Zariski-closed subset in $\mathbb{C}^{n+1}$

Let $n \in \mathbb{N}$, $f\in \mathbb{C}[X_1,\dots,X_n]$, and $D(f):=\{x=(x_1,\dots,x_n)\in \mathbb{C^n}|f(x)\neq 0\}.$ I want to show that there is a injective $\Phi: D(f) \rightarrow ...
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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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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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38 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 ...
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Find $x$ for which the rank is as minimal/maximal as possible

Find an $x$ in $\Bbb R$ for which rank of the matrix $$A=\begin{bmatrix}1 & 1&1&1 \\1 & -1&-1&1\\1 &-3 &-3 &x \end{bmatrix}$$ is as minimal/maximal as possible. I ...
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Derivation of the adjoint of a matrix

Let $V, W$ be vector spaces over any field $F$. A transformation $T:V \rightarrow W$, gives rise to the adjoint $V^* \leftarrow W^*:T^*$ of the dual spaces via: $$ T^*(f)(\cdot) = f\circ T(\cdot) $$ ...
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Clarifications regarding matrix transformations.

I have an equation which looks like this: Pos1 * L1 * X * L2 = Pos2 * R1 Where Pos1 and Pos2 are vectors. L1,X,L2 and R1 are matrices. I have to find the value for the matrix X. Please let me know ...
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Matrix problem in Mixed Regression

the background is $y= X \beta +e$ y=n*1 X=n*p $\beta=p*1$ e=n*1 take singular value decomposition of X $X=P \Delta Q$ $\beta=QKP'y$ K is a digaonal matrix and depending on its form can represent ...
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Binary Polynomial Factoring

I just need confirmation that I've done my math right. If $a(x) = x^4 + x^3 + x + 1$ and $b(x) = x^2 + x + 1$ are binary polynomials, find binary polynomials s(x) and r(x) such that $x^4 + x^3 + x + ...
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Primitive elements of GF(8)

I'm trying to find the primitive elements of GF(8), the minimal polynomials of all elements of GF(8) and their roots, and calculate the powers of α^i for x^3 + x + 1. If i did my math correct, I ...
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2answers
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What is the computational cost of reduced row echelon and finding the null space?

I'm taking computational linear algebra, and haven't been able to find too much information about the computational cost (in terms of m=rows and n=cols) of these two routines: Reduced Row Echelon ...
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Diagonalise without finding eigenvalues

I am asked to find the Jordan normal form (in this case, diagonalise) the $n\times n$ matrix $M$ defined: $$M_{ij}=1+\delta_{ij}\,x$$ I am then asked to deduce the minimal polynomial, eigenvalues and ...
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24 views

Linear algebra determinant-area relation question

I have an exercise where I am transforming a unit circle into an ellipse by some transformation $A$. Is it true that after the transformation the ellipse will have an area $\pi\cdot\mathrm{det}(A)$? ...
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Equation for a plane perpendicular to a line through two given points

The following type of question is quite popular with examiners at the institution where I study. Find an equation of the plane containing the point $(0, 1, 1)$ and perpendicular to the line passing ...
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Generator matrix of a Reed-Muller code [duplicate]

I need to find a generator matrix (2,4) of the Reed-Muller code (2,4), the dimension of R(2,4) and the minimum distance of R(2,4). I know that R(r,m) of order r, then length: n^m, dimension k = 1 + ...
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1answer
21 views

Complex matrix similar to a matrix with identical diagonal entries

Let $A$ be a complex matrix. Show that it is similar to a matrix with identical diagonal entries. I do have some sense, but could not prove it.
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Why inner product < , > on $C^n$ must satisfy the parallelogram law?

Why must norm induced by an inner product < , > on $C^n$ satisfy the parallelogram law? I know that there is a proof using $||v|| = \sqrt{(< v, v>)} $. But my concern is that why it still ...
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Positive definiteness of block matrices

I really appreciate if anyone can help me regarding my problem. I have a matrix in the format $M = \left[ {\begin{array}{*{20}{c}} {\delta I}&A\\ {{A^T}}&kA \end{array}} \right]$ where $A$ is ...