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 […]

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: […]

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.

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 […]

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 […]

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 […]

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 […]

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 […]

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 […]

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 […]

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