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 ...

learn more… | top users | synonyms (1)

0
votes
1answer
25 views

Given $v \in C^n$ that $u^Hu = 1$, and $D = iuu^H$ find all eigenvalues of $D$

Given $v \in C^n$ that $u^Hu = 1$, and $D = iuu^H$ find all eigenvalues of $D$ Well, I believe that $D$ is composed of orthonormal vectors, because of $u^Hu = 1$. Which means I believe that all ...
0
votes
0answers
16 views

What is the point of solving a system of linear equations using back-substitution (as opposed to reduced echelon form)

In lecture the other day, my professor offhandedly mentioned the existence of a process called back-substitution a way in which a computer program would solve a system of linear equations rather ...
0
votes
1answer
17 views

Find the spectrum of graph

Find the spectrum for the following graph by calcuation The spectrum of a graph $G$ is a list of the eigenvalues and the multiplicities of the eigenvalues of the adjacency of matrix $A$ of $G$. ...
0
votes
1answer
23 views

vector space homomorphism for $Map(\mathbb{F}_{5} , \mathbb{F}_{5})$

I'm currently stuck at a mathematical problem and I really don't know where to start.. Since I'm not an expert in Algebra over finite fields... It goes "Define a $\mathbb{F}_{5}$-vector space ...
1
vote
3answers
44 views

Does a repeated eigenvalue always mean that there is an eigenplane under the transformation matrix?

If you have a 3x3 matrix, if you find that it has repeated eigenvalues, does this mean that there is an invariant plane (or plane of invariant points if eigenvalue=1)? I always thought that there was ...
1
vote
4answers
53 views

Basis in Linear Algebra [on hold]

I am taking an introductory linear algebra course, and I am stuck on this problem: Explain why the set $W= \{(a,b,c)\ |\ a+b+c=0\}$ is a subspace of $\mathbb R^3$. After, find a basis for the ...
1
vote
1answer
34 views

Orthogonal matrix $Q$ such that $\forall x\leq 0$, $Qx\geq 0$

What are the orthogonal matrices $Q$ such that for all vectors $x\leq 0$, $Qx\geq 0$? The inequality is to be understood component-wise. In dimension 1, the only possibility is $Q=[-1]$, which is a ...
2
votes
0answers
22 views

Constraints on a Chebyshev series representation of a CDF

My question is about deriving constraints for coefficients of a Chebyshev series which represents a CDF. Let $F(x)$ be the cumulative distribution function for $x\in [-1,1]$. Accordingly we know ...
0
votes
1answer
44 views

Determinant of orthogonal matrix

If $A$ is orthogonal. how do I show that $\det(A-2I)\not=0$. I tried writing $A-2I=A-2AA^T=A(I-2A^T)=A(A^TA-2A^T)=AA^T(A-2I)$ but it seems that I am just doing loops after loops.
2
votes
3answers
46 views

Why do we have the following implication if $\phi$ is injective

If $\phi: G \rightarrow H$ is a homorphism, and if $\phi$ is injective, why do we have the following: $\phi(g) = e_h \implies g=e_g$
0
votes
0answers
9 views

incremental knapsack

Is there a way to compute the knapsack problem incrementally? Any approximation algorithm? I am trying to solve the problem in the following scenario. Let D be my data set which is not ordered and ...
0
votes
0answers
8 views

Aligning matrices, normalization. Calculating coefficients.

So as a pre-task for my upcoming exam this is one of the rehearsal assignments. I can't wrap my head around this one at all, haven't seen anything like it earlier, and I can't seem to find any ...
0
votes
0answers
19 views

Schur complement of a matrix $A$

Let $A\in\mathbb{R}^{n\times n}$ and its inverse be partitioned $$A = \begin{pmatrix} A_{11} & A_{12}\\ A_{21} & A_{22}\\ \end{pmatrix},\:\: A^{-1} = \begin{pmatrix} \tilde{A_{22}} & ...
0
votes
0answers
18 views

Is it possible to represent {$0, ±m, ±2m, ±3m, \ldots$} in an augmented matrix? [on hold]

An augmented matrix of a system consists of the coefficient matrix with an added column containing the constants from the right sides of the equations. Source: Linear Algebra and Its Applications, ...
0
votes
3answers
42 views

Find intersection point of two straight lines

I want to find the intersection point of two lines where, one of the lines is parallel to y axis. I know we can find the intersection point of two line by solving the equation $y=m(x-P_x)+P_y$ where m ...
0
votes
1answer
13 views

Computing $PAQ = LU$ using Gaussian elimination with complete pivoting

Suppose $PAQ = LU$ is computed via Gaussian elimination with complete pivoting. Show that there is no element in $e_i^{T}U$ i.e., row $i$ of $U$, whose magnitude is larger than $|\mu_{ii}| = ...
2
votes
2answers
48 views

Let $T: V \rightarrow V$ be a linear map, where $nullity(T) = dim(V) - 1$. Prove there is a $\lambda$ such that $T^{2}(v) = \lambda T(v)$.

Let $T: V \rightarrow V$ be a linear map, where $nullity(T) = dim(V) - 1$. Let $w$ be a vector from the image of $T$. If $T(w) \neq 0$, prove there is a non-zero number $\lambda$ such that $T^{2}(w) ...
1
vote
1answer
58 views

Prove that this $10 \times 10$ matrix is diagonalizable. [on hold]

Suppose that $A$ is an non-invertible $10\times10$ real matrix, and that $\mathrm{rank}(A-3I)=7$ , $\mathrm{rank}(A-I)=4$. How do I prove that $A$ is diagonalizable?
0
votes
1answer
26 views

Counterexample of Converse of “rank(PA)=rank(A) if P is invertible”

studying linear algebra , i got a theorem, " Let A be an m x n . If P and Q are invertible m x m and n x n matrices, respectively, then (a) rank(AQ) = rank(A) (b) rank(PA) = rank(A) i know how ...
0
votes
1answer
46 views

Does $AA^T = A^TA$ imply that A is normal?

A is $n\times n$ matrix over complex numbers. Does $AA^T = A^TA$ imply that A is normal? If not what will be a counterexample?
0
votes
0answers
13 views

How to find the irreducible factorisation over Z

So the question is to find the irreducible factorisation of 1-11$\sqrt-2$ over Z[$\sqrt-2$]. I have only been shown how to find this if we have already found the gcd of this with another value, how ...
0
votes
1answer
25 views

Finding span of intersection of two vector subspaces

I was trying to follow this answer, but as the comment to that answer suggests, there's a problem with dimensions, and that's exactly where I'm stuck. More concretely, I have subspaces $U$ and $W$, ...
0
votes
1answer
12 views

Space generated by vectors

I have a doubt: can you say, for sure, that every space generated by two linear independente vectors with two components generate $\mathbb{R^2}$? For example: $L$ {$(1,1),(0,2)$} = $\mathbb{R^2}$ ...
1
vote
0answers
49 views

4 points in 3-d space (one known and three unknown)

Problem in 3-d space. We have four points: $P_0$ where we know coordinates $(0,0,0)$ and $P_1, P_2, P_3$ where coordinates are unknown. However we know distances between $P_1, P_2, P_3$ (let's name ...
-1
votes
0answers
24 views

How to use Euler's formula to get the following identity

I'm reading a textbook and in the chapter on Euler's formula it is said that it's very useful for deriving all sorts of trigonometric identities, and the example given is: Where ||zθ|| = 1 I've ...
0
votes
0answers
30 views

proof about rows and columns in linear algebra

I am in an introductory linear algebra course, and I really need help on this question: Prove that if $P$ and $Q$ are $n\times n$ matrices such that at least one of them has rows that don't span ...
-3
votes
0answers
23 views

Linear Transformation from alpha to beta [on hold]

Hello I am in my Calculus 4 class and I am studying for the final and one thing I've not ever been able to understand is how to do that matrix representation. so I'm working on the practice final ...
1
vote
1answer
30 views

Finding all orthogonal matrices commuting with a positive-definite matrix

Given $M$ a symmetric positive-definite matrix, I'd like to characterise the orthogonal matrices $Q$ commuting with $M$: $MQ=QM$. $Q$ and $M$ commute if and only if they are simultaneously ...
2
votes
1answer
21 views

Using inverse of transpose matrix to cancel out terms?

I am trying to solve the matrix equation $A = B^TC$ for $C$, where $A$, $B$, and $C$ are all non-square matrices. I know that I need to utilize $M^TM$ in order to take the inverse. I'm just not sure ...
-1
votes
2answers
38 views

For what value of k does the following system of linear equations have infinitely many solutions?

I've been struggling for hours trying to solve this: For what value of k does the following system of linear equations have infinitely many solutions? $$x+y+kz=3$$ $$x+ky+z=-7$$ $$kx+y+z=4$$
1
vote
1answer
33 views

Kernel and Image of an integral.

Im struggling to answer a question where $F: P_{2}(\mathbb{R}) \rightarrow P_{3}(\mathbb{R}) $ $$F(f)(x)=\int^{x+1}_{2-x} (1-t)f(t) dt$$ So to find the Kernel do i set the integral equal to 0 and ...
0
votes
1answer
29 views

Kernel of a polynomial with matrix, $ker(p(A))$

Let $A\in Mat(3,3,\mathbb R)$ a matrix and $\chi_A(x)=p_1(x)\cdot p_2(x)$ the characteristic polynomial. Evaluate $ker(p_1(A))$.$$A=\begin{pmatrix} 0 & 0 & 2 \\ 1 & 0 & 1\\ 0 & ...
0
votes
1answer
22 views

Gauss-Jordan elimination/matrix

Hello guys i got a problem from university and i cant seem to find the answer This is the problem : ka+b+c+d=1 a+kb+c+d=1 a+b+kc+d=1 ...
2
votes
2answers
48 views

Solution of $A^\top M A=M$ with $M$ positive-definite

I am trying to find all matrices $A$ such that for all positive-definite matrices $M$, $A^\top M A=M$. $I$ and $-I$ are obvious solutions. I can't find out it there are other such matrices and if so, ...
0
votes
2answers
21 views

Show that this MC is ergodic?

Suppose I have a Markov Chain with States, $S = {1,2,3,4}$ and a PTM given by $P =$ $\begin{pmatrix} .25 & .25 & .25 & .25 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 ...
1
vote
2answers
69 views

Prove every finite lattice has a greatest element - without induction

I have to prove that every finite lattice (L, ≤) has a greatest element. I have seen a lot of proofs proving this by using induction, however, I have to prove it without induction since our ...
0
votes
2answers
30 views

Nullity of linear transformation

I'm struggling to find the nullity $N(T)$ of the following linear transformation (in the canonical basis of $\mathbb{R^{2\times2}}$ $ M = \begin{bmatrix} 0 & 0 & 0 & 0 ...
-1
votes
1answer
66 views

Eigenvalues of matrix of order $n+1$

How to find eigenvalues of following matrix? $A=\begin{bmatrix} n & -1 & -1 & \cdots & -1 \\ -1 & 1 & 0 & \cdots & 0 \\ -1 & 0 & 1 & \cdots & 0 \\ ...
0
votes
0answers
11 views

Convert equation of plane to parametric form , vector form and cartesian form

Find equation of a plane passing through point A(1,2,3), B(3,–1,4), and C(5,1,–4) in: a. Vector form b. Parametric form c. Cartesian form If its equation of line i understand that but for ...
0
votes
1answer
18 views

Matlab algorithm for non-orthogonal diagonalization of symmetric matrices

I need to find a basis in which the symmetric bilinear form given by the n x n symmetric matrix which has 2's along the diagonal and 1's everywhere else becomes the identity. That is, if S denotes ...
1
vote
1answer
83 views

$\mathbb{Z}[x]$ doesn't have principal maximal ideals [on hold]

Prove that $\mathbb{Z}[x]$ doesn't have principal maximal ideals. Please, I need help with this problem. Thanks!
0
votes
2answers
31 views

Relation between eigenvectors of matrix $X^TX$ and $XX^T$

I found a surprising property of the eigenvectors of the matrix $A = X^T X$ and $B = XX^T$ experimentally. Let $X$ be $n \times d$ with $n > d$. Then $A$ and $B$ are psd matrices. The eigenvalues ...
0
votes
1answer
20 views

Basis for the space of 4*4 hermitian matrices with specific anti-commutation properties

The space of 2*2 hermitian matrices can be spanned using the basis involving identity and the three pauli matrices. Here, the pauli matrices have specific properties like: When squared they give ...
2
votes
3answers
32 views

A question about linear combination

The question is to show Given a non-zero vector u and a set of non-zero vectors $D=\{v_1,v_2,…,v_n\}$, show that $u$ is not a linear combination of $D$ if $u⋅v_i=0$ for all of $i=1,2,…,n$. It is ...
2
votes
2answers
36 views

Check if my trajectory colliding another objects

I'm new to Math.stackechange and i'm a programmer not a mathematician :-(. I'm solving problem in 3D engine for a computer game. But this time i need to do calculations on server side, ...
0
votes
0answers
24 views

Intersecting 3 Parametric lines

Given $[x,y,z] = [x0,y0,z0] + t[a0,b0,c0]$ $[x,y,z] = [x1,y1,z1] + s[a1,b1,c1]$ $[x,y,z] = [x2,y2,z2] + v[a2,b2,c2]$ How can I solve for the best intersection ...
1
vote
1answer
82 views

How to deduce the formula for quadratic form?

I almost every book about quadratic form we can see it described as following function: $$ f(x) = \frac{1}{2}x^T A x - b^Tx + c $$ My question is: How can we deduce this formula? I understand, ...
0
votes
1answer
23 views

Easiest way to compute singular values of matrix

Let $A\in GL_2(\mathbb{R})$ be an invertible matrix. I know $A$ has a singular value decomposition $A=U\Sigma V^T$ where $U$ and $V$ are orthogonal matrices and $\Sigma$ is diagonal. I call "singular ...
1
vote
1answer
14 views

how to find the pivot/axis and angle that move one coordinates space to another?

I am writing a plugin for a 3d modeler, and I am stuck. For my plugin, I need to get the axis and the angle used for rotating a 3d object. But I only get the coordinates (~ 3dmatrices) of the objects ...