Solution to Linear Algebra Hoffman & Kunze Chapter 9.2 Exercise 9.2.2 Solution: It is not a form! I think it should be $$f((x_1,y_1),(x_2,y_2))=x_1x_2+y_1y_2.$$Then the matrices are\[\begin{pmatrix}1 & 0\\ 0 &…
Solution to Linear Algebra Hoffman & Kunze Chapter 1.3 Exercise 1.3.8 Solution: (a) In this case the system of equations is \begin{alignat*}{1} 0\cdot x_1 + 0\cdot x_2 &= 0\\ 0\cdot…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.6 Solution: (1) By definition, $E_{i,j}A = [c_{p,q}]$, where $$c_{p,q} = \sum_{k=1}^n e_{p,k}a_{k,q}.$$ Note that if $p…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.9 Prove that the binary operation of matrix multiplication of $2 \times 2$ matrices over $\mathbb{R}$ is…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 0.1 Exercise 0.1.4 Denote by $\mathcal{A}$ the set of all $2\times 2$ matrices with real number entries. Let $M…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 0.1 Exercise 0.1.3 Denote by $\mathcal{A}$ the set of all $2\times 2$ matrices with real number entries. Let $M…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 0.1 Exercise 0.1.2 Denote by $\mathcal{A}$ the set of all $2\times 2$ matrices with real number entries. Let $M…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 0.1 Exercise 0.1.1 Denote by $\mathcal{A}$ the set of all $2\times 2$ matrices with real number entries. Let $M…
Multiply a) $\begin{pmatrix} 1 & 2 & 3\\ 4 &5 & 6\end{pmatrix}\begin{pmatrix} 1 \\ 3\\ 2 \end{pmatrix}$; b) $\begin{pmatrix} 1 & 2 \\ 0 & 1\\ 2 & 0\end{pmatrix}\begin{pmatrix} 1…