Articles of linear algebra

Eigenvector and its corresponding eigenvalue

For the following square matrix: $$ \left( \begin{array}{ccc} 3 & 0 & 1 \\ -4 & 1 & 2 \\ -6 & 0 & -2 \end{array} \right)$$ Decide which, if any, of the following vectors are eigenvectors of that matrix and give the corresponding eigenvalue. $ \left( \begin{array}{ccc} 2 \\ 2 \\ -1 \end{array} \right)$ […]

trace of the matrix $I + M + M^2$ is

Let $ \alpha = e^{\frac{2\pi \iota}{5}}$ and the matrix $$ M= \begin{pmatrix}1 & \alpha & \alpha^2 & \alpha^3 & \alpha^4\\ 0 & \alpha & \alpha^2 & \alpha^3 & \alpha^4\\ 0 & 0 & \alpha^2 & \alpha^3 & \alpha^4 \\ 0 & 0 & 0 & \alpha^3 & \alpha^4\\ 0 & 0 & 0 & 0 […]

What is an application of the dual space?

Does somebody know any application of the dual space in physics, chemistry, biology, computer science or economics? (I would like to add that to the german wikipedia article about the dual space.)

Do $T$-invariant subspaces necessarily have a $T$-invariant complement?

Suppose $T$ is a linear operator on some vector space $V$, and suppose $U$ is a $T$-invariant subspace of $V$. Does there necessarily exist a complement (a subspace $U^c$ such that $V=U\oplus U^c$) in $V$ which is also $T$-invariant? I’m curious because I’m wondering if, given such $U$, it is always possible to decompose the […]

Complex square matrices. Difficult proof.

$det(I+A\cdot\bar{A}) \ge 0$ Is it possible to prove the inequality is true for all complex square matrices $A$ where $I$ is the identity matrix and $\bar{A}$ is the complex conjugated matrix.

Fourier Transform: Understanding change of basis property with ideas from linear algebra

The notion of Fourier transform was always a little bit mysterious to me and recently I was introduced to functional analysis. I am a beginner in this field but still I am almost seeing that the Fourier transform can be viewed as a change of basis in a space of functions. I read the following […]

If $X$ is linearly independent, then $T(X)$ is also linearly independent?

Here is my homework question, Let $V$ and $W$ be vector spaces over a field $F$ and let $T$ from $V$ to $W$ be an isomorphism. Let $X$={$v_{1},v_{2},…,v_{n}$} be a subset of V, and recall that T(X)={$T(v_{1}),T(v_{2}),…,T(v_{n})$}. a. Prove that if $X$ is linearly independent, then $T(X)$ is also linearly independent. b. Prove that if […]

Can $A, B$ fail to commute if $e^A=e^B=e^{A+B}=id$?

Consider the real $n \times n$-matrices $A$ and $B$. Can $A, B$ fail to commute if $e^A=e^B=e^{A+B}=id$ ?

Is there an example of an orthogonal matrix which is not diagonalizable

Based off the theory I can’t see any reason that an example would not exist. Specifically, the fact that A matrix is orthogonal only implies that the possible eigenvalues are $\pm 1$. However, we don’t know anything about the sizes of the eigenspaces. Nonetheless, it is not hard to show that a 2×2 orthogonal matrix […]

Finding points on two linear lines which are a particular distance apart

I have two linear, skew, 3D lines, and I was wondering how I could find a points on each of the lines whereby the distance between the two points are a particular distance apart? I’m not after the points where the lines are the closest distance apart, nor the furthest distance apart, nor the point […]