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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How to calculate the inversion of a triangular matrix

Now I want to write a piece of code to calculate the inversion of a triangular matrix which do it in parallel. I know that the equation of the triangular matrix's inversion is like this: But I ...
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143 views

Properties shared by similar and unitary similar matrices.

We know that matrices $A$ and $B$ are similar if there exists an invertible matrix $P$ such that $A=PBP^{-1}$ and they are unitarily similar if $P$ is unitary ($PP^*=P^*P=I$). I want to know : What ...
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48 views

linear algebra texts suggestions

I am looking for a textbook about linear algebra. I want one with a pure math/algebraic approach and not one with a geometric or a applied/numerical approach. Do you have any suggestions? Thank you
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75 views

If $A $ is a square matrix of size $n$ with complex entries such that $Tr(A^k)=0 , \forall k \ge 1$ , then is it true that $A$ is nilpotent ? [duplicate]

If $A$ is a square matrix of size $n$ with complex entries and is nilpotent , then I can show that all the eigenvalues of $A^k$ , for any $k$ , is $0$ , so $Tr(A^k)=0 , \forall k \ge 1$ . Now ...
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18 views

Extra Operation required for Smith Normal Form over PID-Theoretical Justification

Why does one need an extra operation for performing smith normal form over a PID? One might suspect and say that it is because of the lack of Euclidean algorithm or just say that we need the ...
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29 views

Function's Transformation's Roots.

Given $f(x)=x^3+3x^2+6x+2\sin x$ Transformed into: $$g(x)=\frac{1}{x-f(1)}+\frac{1}{x-f(2)}+\frac{3}{x-f(3)}$$ What can be said about the number of roots of this transformation. ...
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49 views

Linear Equations solution for only 1 known RHS constant

I have 2 variables x and y, and 4 linear equations as follows: $$\begin{equation} C1 \,x =a,\\ C2\,x + C3\,y =b,\\ C4\,x + C5\,y =c,\\ C6\,x + C7\,y =d.\\ \end{equation} $$ where $ C1, ...
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40 views

Square coefficient matrix, matrix transpose, and solvability of the corresponding system of equations

Let $\mathbb{F}$ be a field and $n \geq 2$. I would like to prove that, for every $n \times n$ matrix $A$ over $\mathbb{F}$, there is a $b \in \mathbb{F}^{n}$ such that $Ax = b$ is unsolvable if and ...
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30 views

Reference to line parametrization

Defining two lines in space, $\mathbb{R}^3$, as: $l_1: \textbf{a}_1+\lambda_1\textbf{b}_1$ $l_2: \textbf{a}_2+\lambda_2\textbf{b}_2$ The line to line intersection condition is: $\textbf{b}_1\cdot ...
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24 views

Existence of a particular vector in kernal of $d \exp_x$

I want to prove $(d\exp_x)_{p}$ is singular iff there exists a normal Jacobi field $U(t)$ along $\gamma(t)=\exp_x(tp)$ not identically zero such that $U(0)=U(1)=0$. I have question about ...
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32 views

Determinant of a Linear Transformation 2

Find the determinant of the linear transformation $$T(M) = \begin{bmatrix} 2 & 3 \\ 0 & 4 \end{bmatrix}M$$ from the space $V$ of upper triangular $2 \times2$ ...
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47 views

This proof from the textbook that the number of columns of a matrix equals its rank plus its nullity does not make any sense…

START PROOF "Let A be an m $\times$ n matrix of rank r. Because A has rank r, you know it is row-equivalent to a reduced row-echelon matrix with nonzero rows. No generality is lost by assuming that ...
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46 views

Determining diagonalizability of a matrix containing complex enteries

$$A=\left[\begin{matrix}3-8i&-11+7i\\-1-4i&-2+6i\end{matrix}\right]$$ I've determined the $tr(A) = 1-2i$, and the $det(A)=3-3i$. From here I should be able to use the characteristic equation ...
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33 views

Prove a matrix is Hermitian, if its eigenvalues are real and satisfy an orthogonality relation

Prove a matrix is Hermitian, if: (a) Its eigenvalues are real, and (b) the eigenvectors satisfy $ r_{i}^\dagger r_{j} = \delta_{ij} = \left<r_{i}|r_{j}\right> $ I can see this is the reverse ...
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59 views

Questions on the formula for 2x2 inverse

Hi I was working on inverting 2by2 matrix in general form by using a,b,c,d. I know the formula (which is below) but I have questions in the process of getting the formula. 1) To get rid of the ...
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36 views

Finding linear dependance of a set of functions

where the set $B = \{1+2x+2x^2-x^3,3+2x+x^2+x^3,2x^2+2x^3\}$, how can I show they are linearly independent? Could I set the three vectors, u, v, w, into a coefficient matrix and find it's ...
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37 views

Question about projecting vector onto Subspace

Consider subspace $S \in R^4$, spanned by the vectors: $v_1 = (1,0,-1,1)^T$ and $v_2 = (1,1,1,1)^T$. Let $v = (1,-1,-1,3)^T$. I want to find the projection of $v$ onto $S$; that is, find the ...
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27 views

How to show a set of vectors are a basic for a given plane

To determine if a set, B, of the vectors, u, and v for a basis for the plane, W. let u=(1,2,-1), and v=(1,1,1), W =-3x+2y+z=0 I was able to determine the two vectors, w[1], and w[2], from s and t ...
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33 views

Finding the number of isomorphisms which exist between $\mathbb{Z}_{3}^{2}$ and $\mathbb{Z}_{3}^{2}$

I'm just not sure how to go about this, could you help me to find the method of doing this? Suppose we have a field $\mathbb{Z}_{3} = \{0,1,2\}$ and we take the vector space $\mathbb{Z}_{3}^{2}$ ...
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113 views

Classification of critical points for plane autonomous system

Okay so I've changed the 2nd order nonlinear ODE $$ x'' = a(x')^2 - ax' -ax $$ where a is a real constant, into $$ x' = y $$ $$ y' = ay^2 -ay - ax $$ I'm asked to verify the critical point (0,0). ...
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165 views

Prove a matrix is not diagonalizable

To show that a matrix is not diagonalizable, I would just have to show that there are no eigenvalues present in the matrix. So, for example, if I want to prove that $$A=\begin{bmatrix} 0 & -1 ...
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148 views

Why do lagrange multipliers have the form $\nabla G$

I was studying some multivariable Calculus and we were covering the topic of Lagrange multipliers. I didn't understand exactly why the equations take the form: $$ \nabla f = \lambda \nabla G $$ ...
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28 views

Find an isomorphic map from the space of polynomials into 2-tuple polynomials

We are told that $F[x] \oplus F[x]$ is the space of 2-tuples of polynomials ($F[x]$ is the set of all polynomials). We must find an isomorphic map $S : F[x] \to F[x] \oplus F[x]$. I am not sure if I ...
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44 views

Proof the isomorphism between symmetric group & subgroup ?

For the symmetric group S(2k) there are two equal subsets A = {1,....,k} and A' = {k+1,.....,2k}. Let L be the subgroup of all permutations r of S(2k) with r(A) = A or r(A)= A' and r(A') = A or r(A') ...
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277 views

How to prove that the matrix $A^k$ approaches $0$ as $k$ approaches infinity

First of all, what does it mean to say an eigenvalue is "less than unity"? I'm not exactly sure what this means. Secondly, how do I show that $\lim_{k\to\infty} A^k=0$ given that all eigenvalues of ...
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47 views

Eigenvalue Bound of Block Matrices

I have the following eigenvalue problem for block matrices A and B \begin{equation} \left[ \begin{array}{ccc} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & ...
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144 views

Determine the coefficient of polynomial det(I + xA)

Given matrix an n-by-n matrix $A$ and its $n$ eigenvalues. How do I determine the coefficient of the term $x^2$ of the polynomial given by $q(x) = \det(I_n + xA)$
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72 views

2D rotation matrix: express sin and cos in terms of the elements and the norm of an arbitrary non-zero vector

2D rotation matrix is used to derive the expressions for sin and cos in terms of $a_1 ,a_2 and ||\vec{a}|| $ with the following given I'm trying to figure out where the negative-sign comes from in ...
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58 views

Find the possible signatures of the bilinear forms

Find the possible signatures of the following bilinear forms: The bilinear form $\phi:\mathbb R^n\times\mathbb R^n\to\mathbb R$ given by $\phi(x,y)=x^Tp(A)y$ where $p(t)=t^2+bt+c$ is a ...
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28 views

Solution to a ODE system using a power series

I'm certain the pattern the system creates is $$ A^kX(0) = \begin{pmatrix}2^k\\1\\2^k\end{pmatrix}\hspace{3pc} $$ Where A is a matrix created by the system and X(0) is a solution vector at t=0 Im ...
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Calculate trend of week

Lets say I have the statistics for the crimes that were committed this week in California: On Monday there were 10 crimes committed. Tuesday: 6 Wednesday: 7 Thursday: 15 And on Friday: 8 Now I ...
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Subspace of $L(V)$

Suppose $V$ is finite dimensional and $E$ is a subspace of $L(V)$ such that $ST\in E$ and $TS \in E$ for all $S \in L(V)$ and $T\in E$. Prove that $E=\{0\}$ or $E =L(V)$. When $E$ is non-trivial, I ...
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43 views

Given endomorphisms, show that every one-dimensional subspace is invariant over these endomorphisms

I'm having trouble with this problem and I was wondering if somebody could assist in solving it. So, we have that $W$ is a finite dimensional vector space over some field. We let a linear map $P$ be ...
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58 views

SVD and transpose of a skinny matrix

Show: If $\mathbf{A}\in\mathbb{R}^{M\times N}$ with $M\geq N$, then there exists a matrix $\mathbf{G}$ with orthonormal rows so that $\mathbf{A}^T=\mathbf{G}\mathbf{A}\mathbf{G}$. I'm pretty lost on ...
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173 views

Prove there exists a matrix that satisfies the equation

Let A be an mxn matrix with rank m. Prove that there exists an nxm matrix B such that AB = $I_{m}$. -Since the rank of A is m, and A is an mxn matrix, the matrix must have full row rank, so there ...
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37 views

Positive definite matrix proof with post

Consider $A\in M_n(\mathbb{R})$ a positive-definite matrix and a matrix $B\in M_{nxp}(\mathbb{R})$, with $n\geq p$ and $rank(B)=p$. Show that $$C=B^tAB$$ is positive definite. I have no idea of how to ...
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27 views

If $H$ is a bilinear form then for every $x$ there exists non-null $y$ with $H(x,y)=0$

Prove or disprove: Suppose $H$ is a bilinear form on a finite dimensional vector space $V$, with $\dim(V)>1$. Then for any $x\in V$ there always exists a non-zero $y\in V$ such that $H(x,y)=0$. ...
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46 views

Isomorphisms: $(Aut(V), \circ) \to (GL(n, \mathbb{R}), \cdot)$ and $(Or(V), \circ) \to (O(n), \cdot)$

Let $V$ be a $n$-dimensional $\mathbb{R}$-vector space. Let $Aut(V)$ be the set of the automorphism on $V$. I have shown that this is a group with respect to the composition of functions. However, I ...
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60 views

Characterizations of positive definiteness of a symmetric matrix of order $2$

Let $$M=\pmatrix{ a& b \\b&c }$$ be a symmetric matrix. In my textbook the following result is stated without proof, but I would like to know why it holds, but I cannot figure out what to do ...
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Some doubts and questions with the trace of a matrix

Let $\text{tr}A$ be the trace of the matrix $A \in M_n(\mathbb{R})$. I realize that $\text{tr}A: M_n(\mathbb{R}) \to \mathbb{R}$ is obviously linear (but how can I write down a formal proof?). ...
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Show that $2^{105} + 3^{105}$ is divisible by $7$

I know that $$\frac{(ak \pm 1)^n}{a}$$ gives remainder $a - 1$ is n is odd or $1$ is n is even. So, I wrote $ 2^{105} + 3^{105}$ as $8^{35} + 27^{35}$ and then as $(7\cdot 1+1)^{35} + (7\cdot ...
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100 views

Proximal operator of spectral norm of a matrix

How can I calculate the proximal operator of spectral norm for any general matrix, $X\in R^{m\times n}$ i.e., $X^* = \arg \min_X ||X||_2 + \frac{1}{2\tau} ||X-Y||_F^2$ I understand that the proximal ...
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32 views

Module isomorphism from $R$ to $R \oplus R$ for a certain ring $R$

My textbook says: Let $R$ denote the set of infinite–by–infinite, row– and column–finite matrices with complex entries. Show that $R \cong R \oplus R$ as $R$–modules. So for $A, B \in R$, I tried ...
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120 views

smallest eigenvalue of rank one matrix minus diagonal

Let $x$ be a $d$-dimensional real vector with $\| x\| = 1$. Define $X := xx^T - \mathrm{diag}(xx^T)$. Is it possible to show that $\lambda_{\mathrm{min}}( X ) \geq - 1/2$? Running a bunch of random ...
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41 views

Describing ideal that vanishes at the variety

We have the following morphism $$\phi(a_1,..a_m;b_1,...,b_n)= \begin{pmatrix} a_1 b_1 & \ldots & a_1 b_n \\ \vdots & \ddots & \vdots \\ a_mb_1 & \ldots & a_m b_n ...
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41 views

How to prove: $\alpha$ and $\gamma$ have the same $T$-annihilator, then so do $f\alpha$ and $f\gamma$

I'm reading Hoffman's "Linear Algebra", in $\S7.2$ he mentioned that Let $f$ be a polynomial, $V\in F^n$, $T\in \mathcal L(V)$, and $\alpha,\gamma \in V$. If $\alpha$ and $\gamma$ have the same ...
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27 views

Finding an orthonormal basis which diagonalizes a linear operator

We are given the linear operator $T: \Re^{3} \rightarrow \Re^{3}$ given by the formula $T(x_1,x_2,x_3) = \frac{1}{25} (7x_1-24x_3, -25x_2, -24x_1-7x_3)$. The first part of the question asks us to ...
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477 views

Find vector perpendicular to two vectors without using cross product

I've got a problem with two vectors A=(1,2,-3) and B=(2,-1,3), where I have to find the coordinates for a vector that is perpendicular to both A and B. I know I can use the cross product method for ...
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2answers
112 views

The point A (4, 3, c) is equidistant from the planes P1 and P2. Calculate the two possible values of c

The point $A (4, 3, c)$ is equidistant from the planes $P_1$ and $P_2$. Calculate the two possible values of $c$. Plane $P_1$ has equation $r\cdot (2,-2,1)=1$ Plane $P_2$ has equation $r\cdot ...
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48 views

When does $x^T (xy^T) y = x^T (x^Ty) y$?

$x$ and $y$ are column vectors. When does $x^T (xy^T) y = x^T (x^Ty) y$? After a few trial and errors, I found that if at least one of $x$ and $y$ is a zero matrix then the equality is true. The ...