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

0
votes
0answers
24 views

Extracting the dual feasible search directions for the primal-dual potential reduction algorithm?

I am trying to implement the 4.4 Primal-dual potential reduction algorithm introduced in M.S Lobo et al.. Here is a screenshot depicts the algorithm flow: As ...
0
votes
0answers
28 views

A question about similarity transformation.

Say $A$ is an $n\times n$ symmetric matrix such that every row (and hence column) has exactly $d<n$ non-zero entries. Does there exist similarity transformations on $A$ which will maintain these ...
0
votes
2answers
20 views

gram matrix determines vectors up to isometry

According to http://mathworld.wolfram.com/GramMatrix.html, the gram matrix determines a set of vectors up to an isometry. I'm trying to prove this statement. More specifically, let $A, B \in ...
1
vote
1answer
38 views

orthogonal projection onto linear space of matrices

Let $M$ be an $n_1 \times n_2$ matrix with rank $r$ and let $M = U\Sigma V^T$ be its SVD. Define the space $T = \mathrm{span}\{\{ u_k y^T : y \in \mathbb{R}^{n_2}, 1 \leq k \leq r\} \cup \{ x v_k^T : ...
0
votes
1answer
40 views

How can I define T in this linear transformation?

For $T: V2\to V2$ $T$ maps each point with polar coordinate $(r,\theta)$ to each point with polar coordinate $(r,2\theta)$ and $T$ maps $0$ onto itself. I let $r= \sqrt{ x^2 + y^2}$ and ...
1
vote
1answer
44 views

Is there a way to do this besides brute force?

$A$ is a $d\times n$ matrix and $\mu>0$. I'm trying to show that $$(AA^T + \mu I)^{-1} A = A(A^T A+\mu I)^{-1}.$$ The only way I've thought about doing this was by the brute force method of ...
1
vote
3answers
17 views

General nilpotent matrix to upper right matrix

Is it possible to explicitly give the basis transform matrix $Y$ for transforming a nilpotent 2-by-2 matrix $A$ to a matrix, whose only nonzero entry is in the upper right corner? $Y^{-1}AY=\left( ...
0
votes
2answers
31 views

Intro Linear Algebra Proofs

Hello I am having some trouble coming up with a solution to some text book problems. "If A is a an invertible n x n matrix, show that AX=B has a unique solution for any n x k matrix B." Im not sure ...
-1
votes
1answer
35 views

Find system of equation with infinite solutons?

$$2x+7y-5z=0\\ 5x-2y+6z=1\\ 7x+5y+z=1$$ Answer should be in the form of (blank,blank,z) where z is any real number.
0
votes
0answers
6 views

Has the degree to which a partial eigensystem of a large sparse matrix approximates the complete eigensystem been determined?

Does anyone know of any studies or results regarding the degree of approximation or the error in estimating the complete spectrum of a large sparse matrix by means of its first $n$ eigenvalues and ...
0
votes
1answer
34 views

What are these formulas that are suppose to be Gram-Schmidt

The formulas are in this picture. My question is what are these formulas used for. I tried using them but they don't work. I'm familiar with Gram-Schmidt but these don't look like GRAM. I got these ...
0
votes
0answers
23 views

reflection(reflection) = rotation

Lel $\alpha$ and $\beta$ be two distinct simple roots in a root system $\Phi$. How to prove that i) $S_{\alpha} S_{\beta}$ is a rotation in $\mathbb{R}\Phi$ ii) Composition of two reflection is a ...
0
votes
3answers
47 views

Linear Algebra - args complex number question

I need to solve this problem : $$z^3-(2+2i)^2=0$$ This is what I did : $$z^3 = (2+2i)^2$$ $$z^3 = 8i$$ The formula for args is : $$\tan(args)=\frac{b}{a}$$ in this case its clear that the args is ...
1
vote
0answers
18 views

Condition that multiplied hermitian matrix stays hermitian

Suppose we are given a hermitian matrix $E \in \mathbb{C}^{n\times n}$. I want to find sufficient conditions on the entries of a real symmetric matrix $M$ (depending on the entries of the given ...
2
votes
2answers
46 views

Would Evaluating a polynomial at uniformly random points outputs random values?

I`m wondering if we evaluate a polynomial on many points picked uniformly at random. Can we say the output values Y's are uniformly at random?
0
votes
0answers
31 views

A matrix equation with real coefficients

The problem is the following: Find $\lambda$ such that $ b^{T}A\left[A^{T}A-\lambda L^{T}L\right]^{-1}L^{T}L\left[A^{T}A-\lambda L^{T}L\right]^{-1}A^{T}b-\delta^{2}<0 $ where ...
0
votes
2answers
47 views

Positive matrices are open

An linear application $A:\mathbb R^n\to \mathbb R^n$ is positive when it is symmetric and besides that $\langle Ax,x\rangle\gt 0$ for every $x\neq 0$ in $\mathbb R^n$. I would like to prove the set of ...
4
votes
1answer
100 views

Novel approaches to linear algebra and geometry

I'll be studying Brannan's Geometry and Lang's Introduction to Linear Algebra for one university course. I would like to know if you can you suggest some books that offer a unique perspective on the ...
0
votes
3answers
30 views

show that dim(L,W) = mn

There are two finitely dimension vector spaces $V$ and $W$. Dimensions are $n$ and $m$ respectively. $$L(V,W)=\{T:V\rightarrow W \;|\; T \;\text{is linear}\}$$ $L(V,W)$ is a vector space with ...
0
votes
0answers
6 views

Growth rate, annualized growth

At the end of 2001 we had 1,000,000 oranges. 55% of those oranges were stale. What would be the annualized growth rate needed to increase non-stale by 75,000 oranges over four years? Show your ...
1
vote
3answers
35 views

Why solving a system of linear equation produces the intersection of the equation

1) $x+y=1$ 2) $-x+y=1$ Geometrically we can visualize the two lines will intersect at $x=0, y=1$. Consider this algebraic solution using Gaussian Elimination, . But why do they be reduced to the ...
0
votes
1answer
40 views

Closed form solution for a 3x3 matrix given some constraints

I would like to know if it's possible to find a closed form solution (even if not unique) for the $3\times3$ rank-deficient matrix M meeting the following constraints (in the equations below, $x,y$ ...
1
vote
3answers
60 views

To prove that the sum of the roots of the characteristic polynomial of a square matrix is equal to the trace of the matrix

How do we prove that the sum of the roots of the characteristic polynomial of a square matrix is equal to the trace of the matrix ? I want a proof which does not use much computation or determinants ; ...
-2
votes
1answer
23 views

What are the x-intercept for the graph below? [on hold]

What are the x-intercept for the graph below?
1
vote
1answer
29 views

Minimize the Frobenius norm of the difference of two matrices with respect to matrix: $\underset{B} {\mathrm{argmin}} \left\| A- B \right\|_F$

The following question is similar to this one, but I think that it is not straightforward to move from one to the other, so please take a look. Otherwise, please let me know and I will delete it. ...
0
votes
1answer
18 views

Find point on a line using its orthogonal projection

How to find a point $\bf{X}$ on a line from its orthogonal projection $\bf{P}$ on another line. Lets say we have vectors $\bf{A}$, $\bf{P}$ which start at $\textbf{0}$, how to find point $\bf{X}$? ...
1
vote
0answers
26 views

Solve linear system with matlab

In my problem, $A$ is a $m \times n$ matrix with $m \geq n$ and $\mathrm{rank}(A)= n$. Let $\Gamma$ be the $(m+n) \times n$ matrix defined by : $$ \Gamma = \begin{bmatrix} A \\ \mathrm{I_{n}} ...
0
votes
1answer
24 views

Minimal polynomial of f restricted to its image

Let $f:V\to V$ be a $F$-linear map, $V$ an $n$-dimensional vector space over $F$, $\operatorname{rank} E=r$, $W=\operatorname{Im} f$, $\tilde f:=f|_W:W\to W$. Let $\mu$ be the minimal polynomial of ...
2
votes
1answer
21 views

Calculate flight distance from one city to another on earth (sphere) [closed]

I've been sitting with this problem, I really cannot get solved. How do you calculate the flight distance from one city to another, given the longitude and latitude of the cities. For instance, can ...
1
vote
1answer
18 views

Conceptual question on eigenspace

I came across this theorem that says: Let T be a linear operator and let $\lambda_1, \lambda_2, ... \lambda_k$ be distinct eigenvalues of T. For each i = 1, 2, ..., k, let $v_i \in E_{\lambda_i}$, ...
1
vote
1answer
33 views

Diagonalising the symmetric Matrix

I need to diagonalise the following symmetric matrix: $$A = \begin{bmatrix} 1 & 2 & -3 \\ 2 & 5 & -4 \\ -3 & -4 & 8 \end{bmatrix}$$ The characteristic ...
0
votes
1answer
48 views

Why is $(BA)^* = A^*B^*$?

If $U, W,V$ are vector spaces and $A: V \to W$ and $B: W \to U$, why is the pullback operator $$(BA)^*\omega = A^* B^* \omega$$ where $\omega$ is an alternating form Source
0
votes
4answers
82 views

Show that the matrix $I+A$ is invertible when $A$ is nilpotent [duplicate]

If $A$ is a nilpotent matrix, then how to show that the matrix $I+A$ is invertible.
0
votes
1answer
19 views

Point Parallel Form Describe Same Line as Point Normal Form

And that's how far I able to get, any suggestion how I can equate both (bold) equation or did I do totally wrong?
0
votes
2answers
29 views

Show that there exist ordered bases $\beta$ and $\gamma$ for V and W, such that T is a diagonal matrix

Let V and W be vector spaces such that dim(V) = dim (W) and let $T: V \to W$ be linear. Show that there exist ordered bases $\beta$ and $\gamma$ for V and W, such that $[T]^{\gamma}_{\beta}$ is a ...
1
vote
0answers
31 views

Let $A\in\operatorname{M}_n(F)$ be a matrix, how to prove $\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}$.

Let $A\in\operatorname{M}_n(F)$. How to prove the following identity. $$\bigcap_{X\in C(A)}C(X)=F[A]=\frac{F[x]}{(m_A(x))}.$$ Here $m_A(x)$ the minimal polynomial of $A$ and $C(A)$ is the ...
0
votes
0answers
14 views

Convertion to Quaternion

Specifications and Data We have a 3D rotation function $R(t)_{3\times 3}$ and function ${K(t)}_{3\times 3}$ a matrix function that gives skew symmetric matrix as out puts. It means it holds the ...
1
vote
3answers
36 views

Inequalities and $x^2$

I would just like to clarify something in regards to inequality and how x^2 would affect it. Why is it that if I have the inequality: $x^2(x+5)(x-6)>0$, for example, I can simply divide out $x^2$? ...
0
votes
3answers
51 views

Proof by Mathematical Induction?

Okay, I always get stuck proving things. I proved that it is true from the first value. I know that now I have to prove that it is true for $n+1$ to show that its true for any $n$. Below I wrote what ...
2
votes
1answer
27 views

Lists vs Sets in linear algebra

I’m currently learning linear algebra from “Linear Algebra Done Right” by Sheldon Axler. The author, in his proofs, makes use of lists of vectors, as opposed to the more conventional usage of sets of ...
2
votes
1answer
69 views

Characteristic polynomial and conjugate matrices question

Suppose you are given a polynomial $p(x)=(x-a)(x-b)$, where $a\neq b$ and $a,b\in\mathbb{Z}$. How many equivalence classes of $\mathbb{Z}$ conjugate $2$ by $2$ matrices having $p$ as their ...
0
votes
2answers
24 views

Determining whether the set of vectors is dependent or independent (with trigs)

I get the idea of determining whether a set of vectors is dependent or independent when it involves polynomials like for example: $\{x^2-1, x^2+1, 4x, 2x-3\} in P$ in this case you can create a ...
0
votes
0answers
15 views

Reviewing some linear algebra and having trouble how to describe sets

For i = 1, 2, 3, . . . , m, let $a_i$· = ($a_{i1}, a_{i2}, . . . , a_{in}$) T ∈ $\mathbb{R}^n$ and $α_i ∈ \mathbb{R}$. (a) For n = 1, 2, 3 and n > 3 with m = n describe the set $S = \{ x ∈ ...
1
vote
1answer
20 views

Standard basis with relative coordinates

Let $L\colon \mathbb{R}_3[t] \to \mathbb{R}_2[t]$ be defined by $L(p)(t) = p'(t) + p(1) + p(2)t + p(3)t^2$. Find the matrix of $L$ relative to the standard bases of $\mathbb{R}_3[t]$ and ...
0
votes
1answer
33 views

Find the equivalence class containing the element

Consider the group G = {1, 3, 5, 7} under multiplication mod 8. Consider the subgroup H= {1,3}. Find the equivalence class containing the element 5 using the relation ~R. I am very stuck on this ...
0
votes
2answers
24 views

Find eigenvalue and eigenvector (linear transformation)

The linear transformation is reflection in the line $y = -4 x$ $(\mathbb{R}^2 \rightarrow \mathbb{R}^2$). It has a eigenvector $[1, -4]$ and a corresponding eigenvalue $1$. Find the other eigenvector ...
0
votes
1answer
29 views

Given these three vectors, find the intersection of the three planes:

Given the three vectors, find the intersection of the three planes: $n_1 = (1,2,3)$ $n_2 = (3,2,1)$ $n_3 = (1,-2,-5)$ What happens if $n_3 = (1,-2,-4)$? Why is this different?
1
vote
1answer
33 views

Why is the infinite dimensional vector space with only finitely many nonvanishing components incomplete?

Define a complex vector space $V$ such that any element $\{a_i\}=(a_1,a_2,\dots)\in V$ has only finitely many components $a_i\ne 0$. The inner product is defined as $$(\{a_i\},\{b_j\})=\sum_i^\infty ...
-1
votes
0answers
25 views

$x$ converges asymptotically to zero

So, I have the following polynomial: $f(x)= \ddot{x} + k_1 \dot{x} + k_2$. What is the condition where $x$ converges asymptotically to zero? Can I get examples of such physical systems that the ...
1
vote
0answers
30 views

What does $p_j(x)$ mean?

I am reading paper The Approximation Power Of Moving Least-squares by David Levin (and few more with same topic), and I have become a bit confused about notations. What confuses me the most is ...