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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Linear algebra: Solving for the coefficients on vectors

I am solving the following system: $$ -\frac{1}{r^2}\begin{bmatrix}\sqrt{\mu}\cos(\theta)\\ \sin(\theta) \end{bmatrix}= ...
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Bounding the off-diagonal entries of a matrix

The Pauli matrices are $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, $X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, $Y = \begin{bmatrix} 0 & -i \\ i & 0 ...
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Show that all multilinear functions are proportional

This was the first problem on the final exam for my undergraduate Linear Algebra class almost 45 years ago. It still haunts me to this day. I think I lost my A to that question! I was caught off guard ...
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for every linear map $\ T:V\to V$ : $\ [T^*]_B=(M^t)^{-1}A^tM\ $ when $\ [T]_B=A$.

Let $V$ be an inner product space of finite dimension over $\mathbb{R}$ and Let $B=\{v_1,...v_n\}$ be a basis of V (not necessarily orthonormal). Let $M\in M_n(\mathbb{R})$ a matrix whose i,j ...
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22 views

Nilpotent Lie Algebras and 2-dimensional Lie Subalgebras

Let be $\mathcal{L}$ a finite-dimensional Lie algebra. How I can prove that if every $2-$dimensional Lie subalgebra of $\mathcal{L}$ is abelian, then $\mathcal{L}$ is nilpotent?
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Question in Introductory Linear Algebra [on hold]

I really need help with this question. I am in an introductory linear algebra course. If you guys could help me, I would really appreciate it. Here is the question: A large apartment building is ...
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4answers
27 views

How can I find the eigenvalues of a $2\times2$ rotation matrix in $\mathbb{R}^2$?

How can I find the eigenvalues of a $2\times2$ rotation matrix in $\mathbb{R}^2$? I tried with $\det(A - aI) = (\cos\phi - a)^2 + \sin^2 \phi = 0$ and I got somehow to $2\cos\phi = a$, and I believe ...
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Tricks for quickly reading off the eigenvalues of a matrix

I noticed that some mathematicians have an uncanny ability to identify the eigenvalues of matrices without doing much in the way of computation. For instance, one might notice that all the rows have ...
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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 ...
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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 ...
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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 $G$. ...
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1answer
21 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 ...
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3answers
37 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 ...
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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 ...
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1answer
31 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 ...
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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 ...
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1answer
38 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.
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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: $\phi(g) = e_h \implies g=e_g$
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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 ...
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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 ...
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17 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}} & ...
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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, ...
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3answers
39 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 ...
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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}| = ...
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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) ...
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1answer
55 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?
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1answer
24 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 ...
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1answer
43 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?
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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 ...
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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 $W$, ...
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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: $L$ {$(1,1),(0,2)$} = $\mathbb{R^2}$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
26 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 ...
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1answer
19 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 ...
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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? $$x+y+kz=3$$ $$x+ky+z=-7$$ $$kx+y+z=4$$
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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 ...
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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 & ...
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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 ...
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2answers
40 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, ...
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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 ...
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2answers
67 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 ...
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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 ...
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1answer
61 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 \\ ...
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0answers
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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 ...
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1answer
17 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 ...
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1answer
80 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!