Articles of linear algebra

Question on Smith normal form and isomorphism

Put $A=\begin{pmatrix} 1 & -5 & 4\\ 1 & -2 & 13\\ -2 & 13 & 7 \end{pmatrix}.$ The smith normal form of this matrix is \begin{pmatrix} 1 & 0 & 0\\ 0 & 3 & 0\\ 0 & 0 & 6 \end{pmatrix} and now I want to find $a , b, c$ $\in$ $\Bbb […]

Prove that if $\ker(T) \subseteq \ker(S)$, then $S = kT$ for some $k\in \mathbb{R}$

Suppose $V$ is a finite dimension linear space with dimension $n$, and that $S,T: V\rightarrow \mathbb{R}$ are linear transformations such that $\ker(T) \subseteq \ker(S)$. What are the possible values of $rank(T)$? Show that $S = kT$ for some $k\in \mathbb{R}$. We have by the rank-nullity theorem that: $$rank(T) + \dim(\ker(T)) = \dim(V)$$ So that: $rank(T) […]

$A^2=cA$ for some $c \neq 0$

Let $A \in \mathbb{C}^{n \times n}$ and $0 \neq c \in \mathbb{C}$ a given constant. Suppose that $A$ has the following property: $$A^2 = cA.$$ Questions. 1) Is there a matrix class for matrices those have this special property? If there is what is the name of it? 2) It is true or not, that […]

Proof that this set is convex

I need a help with prooving that a given set is a convex set: $\{ x \in R^n | Ax \leq b, Cx = d \}$ I know the definition of convexity: $X \in R^n$ is a convex set if $\forall \alpha \in R, 0 \leq\alpha \leq 1$ and $\forall x,y \in X$ holds: $\alpha […]

Does an injective $\mathbb F_9$ vector space homomorphism $\mathbb F_9^3 \to \mathbb F_9^5$ exist?

Does an injective $\mathbb F_9$ vector space homomorphism $\mathbb F_9^3 \to \mathbb F_9^5$ exist? Is it able to solve that task by some technique? If so, how is it working then? I have posted a similiar question here but with a mapping that is not a homomorphism.

Dimension of a vector space is even implies Ker T = Im T.

Let $ V $ be a vector space space of dimension $ n $ even, then there exists a linear map $T : V \rightarrow V $ such that $ Ker T = Im T $ ? The converse is obviously true.

Orthogonal basis for this indefinite symmetric bilinear form

Find an orthogonal basis for the bilinear form over $\mathbb{R}$ given by $(\mathbf{x}, \mathbf{y})\mapsto\mathbf{x}^{t}A\mathbf{y}$ where $A=\begin{bmatrix} 1 & 4 & 4\\ 4 & 4 & 10\\ 4 & 10 & 16 \end{bmatrix}$. I’m not sure if this is as easy as using Gram-Schmidt, or if there is another way. I used Gram-Schmidt and obtained very […]

How to solve simple systems of differential equations

Say we are given a system of differential equations $$ \left[ \begin{array}{c} x’ \\ y’ \end{array} \right] = A\begin{bmatrix} x \\ y \end{bmatrix} $$ Where $A$ is a $2\times 2$ matrix. How can I in general solve the system, and secondly sketch a solution $\left(x(t), y(t) \right)$, in the $(x,y)$-plane? For example, let’s say $$\left[ […]

Hoffman and Kunze, Linear algebra Sec 2.4 Theorem 8

Theorem 8. Suppose $P$ is an $n\times n$ invertible matrix over $F.$ Let $V$ be an $n$-dimensional vector space over $F,$ and let $\scr B$ be an ordered basis of $V.$ Then there is a unique ordered basis $\scr \overline{B}$ of $V$ such that \begin{align} \tag{1}[\alpha]_{\scr B} = P[\alpha]_{\scr \overline{B}} \end{align} for every vector $\alpha […]

A theorem of symmetric positive definite matrix.

Is the following true? Let $g=(g_{ij})\in M(n,\Bbb R)$ be a symmetric positive-definite matrix and let $a=(a_1,\ldots,a_n)\in\Bbb R^n$ be any vector. Then, $$v^Tgv=1\implies (v\cdot a)^2\leq a^Tg^{-1}a,$$ or in other words, $$g_{ij}v^iv^j=1\implies v^iv^ja_ia_j\le g^{ij}a_ia_j,$$ (with Einstein summation). I was trying to prove that the gradient $$grad(f)=g_{ij}\frac{\partial f}{\partial x^i}\frac{\partial }{\partial x^j}$$ on a Riemannian manifold $(M,g)$ is the […]