Articles of linear transformations

Is the matrix of the sum of two linear maps equal to the sum of the matrices of the two maps?

Is the matrix of the sum of two linear maps equal to the sum of the matrices of the two maps? We have bases $\{v_1,\dots,v_n\}$ of $V$ and $\{w_1,\dots,w_m\}$ of $W$. $M(T+S) = M(T) + M(S)$? Whenever $T,S\in \mathcal{L}(V,W)$ What does this look like(in matrix form)? I can’t imagine it for the general case, hence […]

Extending a “linear” map to $\mathrm{span}(S)$

This question stems from the comments and answers of Alex G here. As can be seen in the question or the comments to his answer, we take $V,W$ to be real vector spaces, and say that for $S\subset V$, a function $f:S\rightarrow W$ is “linear”, we’ll say on $S$, if the following two conditions hold: […]

Find a basis for two operators

Let $(E, \langle \cdot, \cdot \rangle)$ be an $n$-dimensional Hilbert space and $A,B \colon E \to E$ linear isomorphisms. Does there exist a basis $\{e_{1},…,e_{n}\}$ of $E$ such that $\mathcal{A}=\{A(e_{1}),…,A(e_{n})\}$ and $\mathcal{B}=\{B(e_{1}),…,B(e_{n})\}$ are orthogonal bases? Hints or solutions are greatly appreciated.

Using Linear Transformations to Solve Differential Equation

Define $T:P_3 -> P_3$ by $T(f)(t) = 2f(t)+(1-t)f'(t)$. a) Show that T is a linear transformation. b) Give the matrix representing T with respect to the “standard basis” {$1,t,t^2,t^3$}. c) Determine ker(T) and Image(T). d) Let $g(t) = 1+2t$. Use your answer from (b) to find a solution to the differential eqn $T(f)=g$. e) What […]

Let $f: V_n \to V_n$ be an endomorphism, prove $\text{dim}(\text{Ker}f \cap \text{Im} f) = r(f) – r(f^2)$

My problem Let $f: V_n \to V_n$ be an endomorphism. Prove that $\text{dim}(\text{Ker}f \cap \text{Im} f) = r(f) – r(f^2)$ where $r$ is the rank of $f$ My solution I am slightly confuse what $\text{dim}(\text{Ker}f \cap \text{Im} f)$ is at all, I know it’s an endomorphism, but I can’t imagine it. I found that if […]

How to find basis of $\ker T$ and $\mathrm{Im} T$ for the linear map $T$?

Find the basis of $\ker T$ and $\mathrm{Im} T$ for the linear map $T:M^{\mathbb R}_{2 \times 2} \to M^{\mathbb R}_{2 \times 2}$ defined as $T(A)=A-A^t$ for all $A \in M^{\mathbb R}_{2 \times 2}$. Let $A=\begin{pmatrix} a&b\\c&d \end{pmatrix}$. Then: $$ T(A)=\begin{pmatrix} a&b\\c&d \end{pmatrix}-\begin{pmatrix} a&c\\b&d \end{pmatrix}=\begin{pmatrix} 0&b-c\\c-b&0 \end{pmatrix}\stackrel{R_2 \gets-1\cdot R_2}{=}\begin{pmatrix} 0&b-c\\b-c&0 \end{pmatrix} $$ In order to find […]

What does this theorem in linear algebra actually mean?

I’ve just began the study of linear transformations, and I’m still trying to grasp the concepts fully. One theorem in my textbook is as follows: Let $V$ and $W$ be vector spaces over $F$, and suppose that $(v_1, v_2, \ldots, v_n)$ is a basis voor $V$. For $w_1, w_2, \ldots, w_n$ in $W$, there exists […]

Rank of $ T_1T_2$

For n$\ne $ m let $ T_1 :R^n \to R^m $ and $ T_2:R^m\to R^n $ be linear transformations s.t $ T_1T_2 $ is bijective. Find rank of $ T_1$ and$ T_2$. I tried by fact that bcoz $ T_1T_2 $ is bijective so rank of $ T_1T_2$ is m.i also find that here n […]

Continuity of infinite $l_p$ matrix

Let $X=(x_{ij})_{i,j=1}^\infty$ be the matrix such that for all $a=(a_n)\in l_p$ and all $k\in\mathbb{N}$ the series $\sum_{n=1}^\infty x_{kn}a_n$ converges in $\mathbb{F}$. We assume that the sequence $Xa=(Xa)_k=\sum_{n=1}^\infty x_{kn}a_n$ is an element of $l_p$. Thus $X:l_p\rightarrow l_p$ is a linear map. Now fix $k,K\geq 1$. We define $X_k^K:l_p\rightarrow \mathbb{F}$ and $X_k:l_p\rightarrow \mathbb{F}$ by $$X_k^Ka=\sum_{n=1}^K x_{kn}a_n$$ and […]

$3\times 3$ Orthogonal Matrices with an Analysis of Eigenvalues

It is said that one can prove that all 3×3 orthogonal matrices correspond to linear operators on $R^3$ of the following types: Rotations about lines through he origin Reflections about planes through the origin A rotation about a line through the origin followed by a reflection about the plane through the origin that is perpendicular […]