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 ...

learn more… | top users | synonyms

1
vote
0answers
7 views

Help me understand Vector Spaces (proving linear spaces)

Please help me understand each part clearly. (i) The reason is its not closed under addition. I know this because I have watched Gilbert Strang's Linear Algebra vedios. But he explains in a ...
0
votes
1answer
17 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 ...
1
vote
3answers
24 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 $x$ ...
0
votes
3answers
36 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 ...
0
votes
0answers
17 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 ...
0
votes
0answers
18 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 ...
1
vote
0answers
14 views

Transient energy growth/nearly parallel eigenvectors and kinetic energy

Here is the question I'm struggling with: I can do part a and b no problem. But I can't figure out what part c is even asking, so I can't figure out where to start solving it. I don't know how to ...
1
vote
2answers
44 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 ...
1
vote
1answer
17 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 ...
1
vote
3answers
51 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 ...
1
vote
0answers
20 views

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 ...
1
vote
0answers
17 views

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 ...
1
vote
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. ...
1
vote
2answers
56 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 ...
0
votes
0answers
22 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 ...
0
votes
1answer
59 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
votes
3answers
45 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 ...
1
vote
2answers
19 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}
7
votes
1answer
42 views

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 ...
1
vote
1answer
17 views

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 ...
3
votes
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 ...
1
vote
0answers
27 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 ...
0
votes
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 ...
0
votes
0answers
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 ...
1
vote
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 ...
0
votes
0answers
20 views

Zariski-open subsace

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 ...
-2
votes
2answers
45 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: $$5-2x-\frac{5-3x}{2}=1$$
1
vote
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.
0
votes
3answers
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 ...
0
votes
2answers
22 views

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 ...
1
vote
0answers
22 views

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: $$ T^*(f)(\cdot) = f\circ T(\cdot) $$ ...
0
votes
0answers
9 views

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 ...
0
votes
0answers
11 views

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 ...
0
votes
0answers
18 views

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 + ...
1
vote
1answer
16 views
0
votes
1answer
12 views

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 ...
0
votes
2answers
14 views

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 ...
0
votes
2answers
36 views

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 ...
0
votes
0answers
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 $\pi\cdot\mathrm{det}(A)$? ...
2
votes
3answers
19 views

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 ...
0
votes
0answers
13 views

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 + ...
1
vote
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.
1
vote
2answers
27 views

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 ...
1
vote
2answers
30 views

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 ...
1
vote
0answers
23 views

Recurrence Derivative

I have a recurrence relation as follows $ \left\{ \begin{array}{ll} R_0=H & \mbox{if } n = 0 \\ R_1(s)=sR_0 \hspace{.1cm} A & \mbox{if }n=1\\ R_{n+2}(s)=\frac{s}{n+2}\{ ...
1
vote
1answer
25 views

Transpose of composition of functions

I am trying to find an alternative proof that $(AB)^t=B^tA^t$. I think it is possible to do by showing that: $(g \circ f)^t=f^t \circ g^t$ where f,g are linear maps between vector spaces. I am ...
0
votes
1answer
12 views

Properties of Hermitian and Positive Definite matrix

Let $A \in \mathbb{C}^{nxn}$ be Hermitian and positive definite. I have to show that $|a_{jk}|^2 < a_{jj}a_{kk}$ $max_{i,j=1,\dots,n}|a_{ij}| = a_{kk}$ for some $k$ with $1\leq k\leq n$ For ...
0
votes
0answers
42 views

Trace of a certain matrix

Let $A$ be a $227 \times 227$ matrix having distinct eigenvalues , with entries from $\mathbb Z_{227}$ , then what is the trace of $A$ ?
0
votes
2answers
43 views

Prove: If $T(u+kv) = T(u) + kT(v)$ then $T$ is linear

I was asked to prove this statement. Prove: If $T(u+kv) = T(u) + kT(v)$ then $T$ is linear It seems to me that for $k=1$ and $u=0$ the statement is proved. Is this correct? Many proofs use this ...
0
votes
2answers
21 views

Is there something called the Reduced Column echleon form?

I recently asked a question where I couldn't find the rank of a matrix. The question is : Problem on Finding the rank from a Matrix which has a variable At the time I believed in the answer, ...