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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38 views

Show that these are the only $\mathbb{R}[x]$-submodule of $\mathbb{R}^2$ [on hold]

Let $T$ be the linear mapping from $\mathbb{R}^2$ to $\mathbb{R}^2$ that is given by a clockwise rotation of $\frac{\pi}{2}$. $T$ makes $\mathbb{R}^2$ an $\mathbb{R}[x]$-module. I want to show ...
2
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0answers
60 views

Are two linear system equivalent? [on hold]

Let $A$ and $M$ be square matrices of size $s$ and $n$ respectively, let $k_i \in\mathbb{R^n}$ be column vectors for all $i=1,\ldots,s$. Denote $K=\left[ \begin{matrix} {{k}_{1}} \\ \vdots ...
-1
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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 ...
1
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1answer
25 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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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 ...
13
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7answers
2k views

Why is a square root not a linear transformation?

The question says: Prove that the function $f(x)=\sqrt{x}$ is not a linear transformation (particularly $\sqrt{1+x^2}≠1+x$) I think that this is because the exponent of $\sqrt{x}$ is $1/2$, ...
2
votes
1answer
35 views

Exponential of a symmetric matrix

Let $A$ be a real, symmetric and positive definite matrix and suppose $B$ is a real symmetric matrix such that $\exp(B) = A$. Is $B$ unique? The solution of my homework sheet says that $B$ is ...
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0answers
12 views

Zero Vector, Closure under addition and Closure under multiplication for a subspace [on hold]

Draw each of the following subsets of the real number plane. Show whether the subspace conditions (Z), (CA), (CM) are satisfied or not. To show that (CM) or (CA) are not satisfied, it suffices to give ...
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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 ...
1
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2answers
38 views

Lower bound on quadratic form

Suppose I have a non-symmetric matrix $A$ and I can prove that $x^T A x = x^T \left(\frac{A+A^T}{2}\right) x>0$ for any $x \ne 0$? Can I then say that $x^T A x \ge \lambda_{\text{min}}(A) \|x\|^2 ...
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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 ...
-1
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2answers
37 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? $$x+y+kz=3$$ $$x+ky+z=-7$$ $$kx+y+z=4$$
2
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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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4answers
725 views

Is there an operation on matrices such that the determinant yields a homomorphism with the additive group of the reals?

It well known that, under standard matrix multiplication $\det(AB) = \det(A)\det(B)$, or in other words, that $\det : \mathbb{R}^{n \times n} \rightarrow \langle\mathbb{R}, * \rangle$ is a monoid ...
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0answers
20 views

Calculating block diagonalization / canonical bases with linear optimization?

In Linear Algebra there are many types of similarity transformations $${\bf A} = {\bf T}^{-1}{\bf DT}$$ Where $\bf D$ is (block-)diagonal. Famous examples include Eigenvalue decompositions, Jordan ...
0
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1answer
28 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
39 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
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1answer
45 views

$\kappa(B^{-1}A)=\kappa (AB^{-1}) = \frac{\lambda_{\max}(B^{-1}A)}{\lambda_{\min}(B^{-1}A)}$

Prove or disprove: if $A,B$ are symmetric positive definite (s.p.d.) matrices then operator 2-norm condition number $\kappa(B^{-1}A)=\kappa (AB^{-1}) = ...
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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!
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1answer
36 views

Linear algebra proof with exchange theorem

Assume the Exchange Theorem and prove the following: Assume the vector space V is finitely generated. Then there is a natural number n such that the length of a linearly independent sequence is less ...
2
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1answer
25 views

Element with maximum magnitude in $A \leq \max(\sigma_{i})$, where $\sigma_{i}$ is singular values of A

Let $A$ be a matrix with real values. Is it true that element with maximum magnitude in $A$ is less than $\max(\sigma_{i})$, where $\sigma_{i}$ is singular values of A? That is, is $$ \max_{ij} ...
1
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1answer
79 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, ...
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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 ...
3
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2answers
133 views

Determinant of specially structured block matrix

How do you compute the determinant of the following $nd \times nd$ block matrix? $$M = \begin{bmatrix}A+B & A & A & \dots & A & A\\ A & A+B & A & \dots & A & ...
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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 ...
0
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1answer
37 views

Create matrix from image

I am strugling with a simple task: Create a matrix $A$ when you know that the image of $A$ has the basis $\langle \{ 1, 4 ,1 \}; \{ 3, 6, 2\} \rangle$ and $A(T)$ (transpose) has the image with basis ...
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1answer
60 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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1answer
55 views

How to compute the projection of a vector on a plane

Can someone check whether my work is correct or not? Compute the projection of $(1,1,1)$ onto the plane that passes through the points $(1,0,-1), (3,7,-3), (-2,-1,2)$. My attempt: Let $u = ...
0
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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 ...
0
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1answer
56 views

Linear Algebra reflection question

Let $T_1$ be a reflection of $\mathbb{R}^3$ about the plane $x = y$ and $T_2$ be a reflection of $\mathbb{R}^3$ about the plane $x = z$. Find standard matrix for the transformation $T_2 \cdot T_1$.
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0answers
8 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 ...
2
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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, ...
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2answers
30 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 ...
2
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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 ...
0
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1answer
19 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 ...
0
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1answer
22 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 ...
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0answers
23 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 ...
14
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7answers
1k views

Determinant of a specially structured matrix ($a$'s on the diagonal, all other entries equal to $b$)

I have the following $n\times n$ matrix: $$A=\begin{bmatrix} a & b & \ldots & b\\ b & a & \ldots & b\\ \vdots & \vdots & \ddots & \vdots\\ b & b & \ldots ...
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1answer
41 views

Express $y = KC^x$ as a linear function

Consider an exponential relationship of the form $y = KC^x$ where $K$ and $C$ are constants. Express the exponential function $y = KC^x$ as a linear function and describe how you would obtain the ...
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1answer
47 views

Dimension of subspace of $\text{End}(\mathbb{R}^5)$

I'm doing a problem which presented me with a basis for some $U\subseteq\mathbb{R}^5$ where $\dim U=3$ (I can give it explicitly if that helps but I do not think it matters). The question is this: ...
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9answers
2k views

A linear operator commuting with all such operators is a scalar multiple of the identity.

The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study. We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a ...
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1answer
27 views

A multiple choice question on span and linearly independent subset of a vector space.

Let $\{v_1,v_2...v_n\}$ be the linearly independent subset of vector space V, where $n\geq 4$. Set $w_{ij}=v_i-v_j$. Let W be the span of set $\{w_{ij}:1\leq i,j\leq n \}$. Then 1.$\{w_{ij}:1\leq ...
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1answer
12 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 ...
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0answers
18 views

Consequence of Cramer's rule and Chiò's condensation [on hold]

enter image description hereHi. I can't understand from where we get this property. I think that is a consequence of Cramer's rule or Chiò's condensation but i haven't found any source that talk about ...
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1answer
17 views

signature of the quadratic form: $f(x,y,z) = xy+yz+xz$

I am asked to find the signature of the following quadratic form: $f(x, y, z) = xy+yz+xz$ I have found that matrix wise, $f(x,y,z)= \begin{bmatrix}x&y&z\end{bmatrix}. ...
7
votes
3answers
6k views

Span of an empty set is the zero vector

I am reading Nering's book on Linear Algebra and in the section on vector spaces he makes the comment, "We also agree that the empty set spans the set consisting of the zero vector alone". Is Nering ...
0
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1answer
18 views

3 linearly- independent vectors

Prove or disprove by counter-example: ${v_1,v_2,v_3}$ linearly-dependent $\Rightarrow$ $ {v_1+v_2,v_1+v_3,v_2+v_3}$ are linearly-dependent. tried to find a counter example and couldn't so I tried to ...
0
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0answers
25 views

Matrix equation with orthogonal matrix

Is there an orthogonal matrix $\mathbf{B}$ such that, for each ${\mathbf{x}} = {\left( {\begin{array}{*{20}{c}} {{x_i}}& \cdots &{{x_K}} \end{array}} \right)^T},{x_i} \geq 0\;\forall i$, : ...
0
votes
1answer
17 views

Proving that the line CR passing through intersection of altitudes AP and BQ is orthogonal to AB

How would you go about solving this? I've tried using projections to prove CO.AB = 0 but haven't made much progress.