Matrix Representation & Linear Mappings Part 2

Published 8/20/2025

Problem 1. (Full Walkthrough)

\text{Let } B=\{(1,0,1),(1,1,0),(0,1,1)\},\quad C=\{(1,1,0),(0,1,1),(1,0,0)\},\quad [F]_{C\leftarrow B}= \begin{bmatrix} 1&2&0\\ 0&1&-1\\ 2&-1&1 \end{bmatrix},\quad v=(3,2,-1). \\[18pt] \textit{Solution.}\\ \text{1) Change } v \text{ from standard to } B\text{-coordinates.}\\[10pt] P_B=\begin{bmatrix} 1&1&0\\ 0&1&1\\ 1&0&1 \end{bmatrix},\qquad P_B^{-1}= \begin{bmatrix} \frac12&-\frac12&\frac12\\[4pt] \frac12&\frac12&-\frac12\\[4pt] -\frac12&\frac12&\frac12 \end{bmatrix}. \\[10pt] [v]_B = P_B^{-1}\,[v]_{\varepsilon} = \begin{bmatrix} \frac12&-\frac12&\frac12\\ \frac12&\frac12&-\frac12\\ -\frac12&\frac12&\frac12 \end{bmatrix} \begin{bmatrix}3\\ 2\\ -1\end{bmatrix} = \begin{bmatrix}0\\ 3\\ -1\end{bmatrix}. \\[18pt] \text{2) Apply the given matrix to get }[F(v)]_C.\\[10pt] [F(v)]_C =[F]_{C\leftarrow B}\,[v]_B = \begin{bmatrix} 1&2&0\\ 0&1&-1\\ 2&-1&1 \end{bmatrix} \begin{bmatrix} 0\\[4pt] 3\\[4pt] -1 \end{bmatrix} = \begin{bmatrix} 6\\[4pt] 4\\[4pt] -4 \end{bmatrix}. \\[18pt] \text{3) Convert }[F(v)]_C\text{ to standard coordinates.}\\[10pt] P_C=\begin{bmatrix} 1&0&1\\ 1&1&0\\ 0&1&0 \end{bmatrix},\qquad F(v)=P_C\,[F(v)]_C. \\[10pt] F(v) = \begin{bmatrix} 1&0&1\\ 1&1&0\\ 0&1&0 \end{bmatrix} \begin{bmatrix} 6\\[4pt] 4\\[4pt] -4 \end{bmatrix} = \begin{bmatrix} 2\\[4pt] 10\\[4pt] 4 \end{bmatrix}. \\[18pt] \boxed{\,F(v)=(2,\,10,\,4)\,} \\[24pt]

Problem 2. (Composition)

\text{Let } B=\{(1,0),(1,1)\},\quad C=\{(2,1),(1,0)\},\quad D=\{(1,2),(0,1)\}\ \text{be bases of }\mathbb{R}^2.\\[12pt] \text{Suppose}\\[6pt] [F]_{C\leftarrow B}=\begin{bmatrix}1&3\\ 0&2\end{bmatrix},\quad [G]_{D\leftarrow C}=\begin{bmatrix}2&1\\ 1&1\end{bmatrix}.\\[12pt] \text{(a) Compute }[G\circ F]_{D\leftarrow B}.\\ \text{(b) For }v=(4,-1)\text{ in standard coordinates, find }(G\circ F)(v)\text{ in standard coordinates.}\\[12pt] \textbf{Answer: }[G\circ F]_{D\leftarrow B}= \begin{bmatrix}2&8\\ 1&5\end{bmatrix},\quad (G\circ F)(4,-1)=(2,\,4).\\ \\[18pt]

Problem 3. Apply Transformation

\text{Let } B=\{(1,2,0),(0,1,1),(1,0,1)\},\quad C=\{(1,0,0),(1,1,0),(1,1,1)\},\\ [F]_{C\leftarrow B}= \begin{bmatrix} 2&0&1\\ -1&1&0\\ 0&2&1 \end{bmatrix},\quad v=(1,3,2). \\[6pt] \text{Compute }F(v)\text{ in standard coordinates.}\\[6pt] \textbf{Answer: }F(v)=\left(\tfrac{19}{3},\,\tfrac{14}{3},\,\tfrac{11}{3}\right).\\ \\[18pt]

Problem 4. Recover Standard Matrix

\text{Let } B=\{(1,1),(0,1)\},\quad C=\{(-1,1),(1,0)\}.\\ \text{Suppose }[F]_{C\leftarrow B}=\begin{bmatrix}0&2\\ 1&1\end{bmatrix}.\\ \text{Compute the standard matrix of }F\text{ (with respect to the standard bases).}\\[6pt] \textbf{Answer: }[F]_{\varepsilon\leftarrow \varepsilon}=\begin{bmatrix}2&-1\\ -2&2\end{bmatrix}.\\ \\[18pt]

Problem 5. Change of Basis

\text{Let } B=\{(1,2),(1,0)\},\quad C=\{(1,1),(0,1)\}.\\ \text{Suppose the standard matrix of }F\text{ is }[F]_{\varepsilon\leftarrow \varepsilon}=\begin{bmatrix}3&1\\ 0&2\end{bmatrix}.\\ \text{(a) Compute }[F]_{C\leftarrow B}.\\ \text{(b) For }v=(2,-1)\text{ in standard coordinates, find }[F(v)]_{C}.\\[6pt] \textbf{Answer: }[F]_{C\leftarrow B}=\begin{bmatrix}5&3\\ -1&-3\end{bmatrix},\qquad [F(v)]_{C}=\begin{bmatrix}5\\ -7\end{bmatrix}.

Problem 6. Matrix Representation Identity

\text{Let }F:V\to U\text{ be linear, }S=\{v_1,\dots,v_m\}\text{ a basis of }V,\text{ and }S'=\{u_1,\dots,u_n\}\text{ a basis of }U.\\ \text{Prove that for every }v\in V,\quad \boxed{\,[F(v)]_{S'}=[F]_{S,S'}\,[v]_S\,}.\\[12pt] \textit{Proof.}\\ \text{Write }[v]_S=\begin{bmatrix}c_1\\ \vdots\\ c_m\end{bmatrix}\text{ so that }v=\sum_{j=1}^m c_j v_j.\\ \text{For each }j,\ \text{expand }F(v_j)\text{ in the basis }S':\quad F(v_j)=\sum_{i=1}^n a_{ij}u_i,\ \text{ so }[F(v_j)]_{S'}=\begin{bmatrix}a_{1j}\\ \vdots\\ a_{nj}\end{bmatrix}.\\ \text{Define the }n\times m\text{ matrix }A=\big[\, [F(v_1)]_{S'}\ \cdots\ [F(v_m)]_{S'} \,\big]=(a_{ij}).\ \text{By definition }A=[F]_{S,S'}.\\[6pt] \text{By linearity of }F,\quad F(v)=\sum_{j=1}^m c_j F(v_j)\ \Rightarrow\ [F(v)]_{S'}=\sum_{j=1}^m c_j [F(v_j)]_{S'}.\\ \text{In matrix form,}\\ \quad [F(v)]_{S'}=A\begin{bmatrix}c_1\\ \vdots\\ c_m\end{bmatrix} =[F]_{S,S'}[v]_S.\\[6pt] \text{This is exactly the desired identity.}\ \square

Problem 7. Change of Basis Problem with Domain and Codomain

\text{Let } B=\{(1,0),(1,1)\},\quad B'=\{(2,1),(1,2)\},\\ C=\{(1,1),(0,1)\},\quad C'=\{(1,0),(1,1)\}.\\ \text{Suppose }[F]_{C\leftarrow B}=\begin{bmatrix}2&1\\ 0&3\end{bmatrix}.\\ \text{(a) Compute }[F]_{C'\leftarrow B'}.\\ \text{(b) For }v=(3,1)\text{ in standard basis },\text{ find }[F(v)]_{C'}.\\[6pt] \textbf{Answer: }[F]_{C'\leftarrow B'}=\begin{bmatrix}-3 & -6 \\ 6 & 6\end{bmatrix},\qquad [F(v)]_{C'}=\begin{bmatrix}-3 \\ 8\end{bmatrix}.

Problem 8. Proof Change-of-Basis Matrix decomposition.

\textbf\quad \text{Prove that } P_{B \leftarrow B'} = M_B^{-1} M_{B'}. \\[6pt] \textit{Proof.} \\ \text{Let } B=\{b_1,\dots,b_n\}, \; B'=\{b'_1,\dots,b'_n\} \text{ be two bases of } \mathbb{R}^n. \\ \text{Define } M_B = [\,b_1 \;\; b_2 \;\; \cdots \;\; b_n\,], \quad M_{B'} = [\,b'_1 \;\; b'_2 \;\; \cdots \;\; b'_n\,]. \\[6pt] \text{By definition, } [v]_B = P_{B \leftarrow B'} [v]_{B'} \;\; \text{for all } v. \\[6pt] \text{In standard coordinates: } v = M_B [v]_B = M_{B'} [v]_{B'}. \\[6pt] \text{Substitute } [v]_B = P_{B \leftarrow B'} [v]_{B'} \text{ into } v=M_B[v]_B: \\[6pt] M_B \, P_{B \leftarrow B'} [v]_{B'} = M_{B'} [v]_{B'}. \\[6pt] \text{Since this holds for all } [v]_{B'}, \;\; M_B \, P_{B \leftarrow B'} = M_{B'}. \\[6pt] \text{Multiply by } M_B^{-1}: \;\; P_{B \leftarrow B'} = M_B^{-1} M_{B'}. \\[10pt] \boxed{P_{B \leftarrow B'} = M_B^{-1} M_{B'}}.

Problem 9. Prove the Proposition

\textbf{Proposition.} \quad \text{For any basis } S=\{s_1,\dots,s_n\} \text{ of }\mathbb{R}^n, \\ \text{and any vector } v\in\mathbb{R}^n, \text{ we have } [v]_{\text{std}} = P_{\text{std}\leftarrow S}[v]_S. \\[6pt] \textit{Proof.} \\ \text{By definition, write } v = c_1 s_1 + \cdots + c_n s_n, \;\; [v]_S = \begin{bmatrix}c_1\\ \vdots \\ c_n\end{bmatrix}. \\[6pt] \text{Form the matrix } M_S = [\,s_1 \;\; s_2 \;\; \cdots \;\; s_n\,], \text{ whose columns are the basis vectors in } S. \\[6pt] \text{Then in standard coordinates we have } v = M_S [v]_S. \\[6pt] \text{Hence the change-of-basis matrix from } S \text{ to the standard basis is } P_{\text{std}\leftarrow S} = M_S. \\[6pt] \text{Therefore } [v]_{\text{std}} = M_S [v]_S = P_{\text{std}\leftarrow S}[v]_S. \\[10pt] \boxed{[v]_{\text{std}} = P_{\text{std}\leftarrow S}[v]_S}.

Problem 10. Change of the Basis with Codomain & Domain - 2

\text{Let } B=\{(1,1),(1,0)\},\quad B'=\{(1,2),(2,1)\},\\ C=\{(0,1),(1,1)\},\quad C'=\{(1,0),(0,1)\}\;(\text{the standard basis}).\\ \text{Suppose }[F]_{C\leftarrow B}=\begin{bmatrix}1&2\\ -1&3\end{bmatrix}.\\ \text{(a) Compute }[F]_{C'\leftarrow B'}.\\ \text{(b) For }v=(2,-1)\text{ in the standard basis},\;\text{ find }[F(v)]_{C'}.\\[6pt] \textbf{Answer: }[F]_{C'\leftarrow B'}=\begin{bmatrix}-5 & 2 \\ -5 & 5\end{bmatrix},\qquad [F(v)]_{C'}=\begin{bmatrix}10 \\ 15\end{bmatrix}.

Problem 11. Canonical Form of a 4→3 Linear Map

\text{Let }F:\mathbb{R}^4\to\mathbb{R}^3\text{ be linear with (standard) matrix } [F]_{\text{std}}= \begin{bmatrix} 1&2&0&1\\ 0&1&1&1\\ 1&3&1&2 \end{bmatrix}. \\[6pt] \text{(a) Find bases }B\text{ of }\mathbb{R}^4\text{ and }C\text{ of }\mathbb{R}^3 \text{ such that }[F]_{C\leftarrow B}= \begin{bmatrix}I_r&0\\ 0&0\end{bmatrix}. \text{ Give one valid pair }(B,C).\\ \text{(b) For }v=(3,0,-1,2)\text{ (in the standard basis), compute }[v]_B\text{ and }[F(v)]_{C}. \\[10pt] \textbf{Answer: }\; r=2,\quad B=\{(1,0,0,0),(0,1,0,0),(2,-1,1,0),(1,-1,0,1)\},\; C=\{(1,0,1),(2,1,3),(0,1,0)\}. \\ [F]_{C\leftarrow B}= \begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&0&0 \end{bmatrix},\qquad [v]_B= \begin{bmatrix} 3\\ 1\\ -1\\ 2 \end{bmatrix},\qquad [F(v)]_{C}= \begin{bmatrix} 3\\ 1\\ 0 \end{bmatrix}.

Problem 12. Canonical Form of a 5→4 Linear Map

\text{Let }F:\mathbb{R}^3\to\mathbb{R}^2\text{ be linear with (standard) matrix } [F]_{\text{std}}= \begin{bmatrix} 1&0&1\\ 0&1&1 \end{bmatrix}. \\[6pt] \text{(a) Produce bases }B\text{ of }\mathbb{R}^3\text{ and }C\text{ of }\mathbb{R}^2 \text{ so that }[F]_{C\leftarrow B}=\begin{bmatrix}I_r&0\end{bmatrix}. \text{ Give one valid pair }(B,C).\\ \text{(b) For }v=(2,-1,3)\text{ (in the standard basis), compute }[v]_B\text{ and }[F(v)]_{C}. \\[10pt] \textbf{Answer: }\; r=2,\; B=\{(1,0,0),(0,1,0),(-1,-1,1)\}, \\ C=\{(1,0),(0,1)\}. \\ [F]_{C\leftarrow B}= \begin{bmatrix} 1&0&0\\ 0&1&0 \end{bmatrix},\qquad [v]_B= \begin{bmatrix} 5\\ 2\\ 3 \end{bmatrix},\qquad [F(v)]_{C}= \begin{bmatrix} 5\\ 2 \end{bmatrix}.

Problem 13

\ \ \text{Let }V,W\text{ be finite-dimensional vector spaces over a field }\mathbb{F},\text{ and let }F:V\to W\text{ be linear. Let }B,B'\text{ be bases of }V\text{ and }C,C'\text{ be bases of }W.\ \text{Denote by }[F]_{C}^{B}\text{ and }[F]_{C'}^{B'}\text{ the matrices of }F\text{ in these bases, and by }[v]_{B},[v]_{B'}\text{ the coordinate columns of }v\in V.\ \text{Prove that } \ker F \text{ is basis-independent, i.e. }\\[4pt] \qquad v\in\ker F\ \Longleftrightarrow\ [F]_{C}^{B}[v]_{B}=0\ \Longleftrightarrow\ [F]_{C'}^{B'}[v]_{B'}=0,\\[2pt] \text{and conclude that }\ker F\subseteq V\text{ and }\dim(\ker F)\text{ do not depend on the chosen bases.}\\[10pt] \textbf{Proof outline.}\\ \text{(1) Let }S\text{ be the change-of-basis matrix on }V\text{ from }B'\text{ to }B,\ \text{so }[v]_{B}=S\,[v]_{B'}\ \text{for all }v\in V,\ \text{and }S\text{ is invertible.}\\ \text{(2) Let }P\text{ be the change-of-basis matrix on }W\text{ from }C\text{ to }C',\ \text{so }[w]_{C'}=P\,[w]_{C}\ \text{for all }w\in W,\ \text{and }P\text{ is invertible.}\\ \text{(3) Show the representation-change formula }[F]_{C'}^{B'}=P\,[F]_{C}^{B}\,S.\ \text{(Apply }[\cdot]_{C'}=P[\cdot]_{C}\text{ to }[F(v)]\text{ and use }[v]_{B}=S[v]_{B'}\text{.)}\\ \text{(4) For any }v\in V,\ \ [F]_{C'}^{B'}[v]_{B'}=P\,[F]_{C}^{B}\,S[v]_{B'}=P\,[F]_{C}^{B}[v]_{B}.\\ \text{Since }P\text{ is invertible, }[F]_{C'}^{B'}[v]_{B'}=0\ \Longleftrightarrow\ [F]_{C}^{B}[v]_{B}=0.\\ \text{(5) But }[F]_{C}^{B}[v]_{B}=[\,F(v)\,]_{C},\ \text{so }[F]_{C}^{B}[v]_{B}=0\ \Longleftrightarrow\ F(v)=0.\ \text{Thus}\\ \qquad \{v\in V:\ [F]_{C}^{B}[v]_{B}=0\}=\{v\in V:\ F(v)=0\}=\{v\in V:\ [F]_{C'}^{B'}[v]_{B'}=0\}=\ker F.\\ \text{(6) Hence }\ker F\text{ is the same subset/subspace of }V\text{ regardless of bases, and its dimension (the nullity) is invariant under any change of basis.}

Matrix Representation & Linear Mappings Part 2

Problem 1. (Full Walkthrough)

Problem 2. (Composition)

Problem 3. Apply Transformation

Problem 4. Recover Standard Matrix

Problem 5. Change of Basis

Problem 6. Matrix Representation Identity

Problem 7. Change of Basis Problem with Domain and Codomain

Problem 8. Proof Change-of-Basis Matrix decomposition.

Problem 9. Prove the Proposition

Problem 10. Change of the Basis with Codomain & Domain - 2

Problem 11. Canonical Form of a 4→3 Linear Map

Problem 12. Canonical Form of a 5→4 Linear Map

Problem 13

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