Exercise 1 Let $ S \subseteq V $ be a subset of vectors in a vector space $ V $. A finite subset $ S' \subseteq S $ is maximally linearly independent in $ S $ if $ S' $ is linearly independent, and for any $ v \in S \setminus S' $ the set $ S' \cup \{v\} $ is not linearly independent. Prove that: (i) $ S' $ is maximally linearly independent in $ S $ if and only if $ S' $ (viewed as a sequence of vectors) is a basis for $ \operatorname{span}(S) $. ...
Exercise 1 Give a basis for the vector space $ M_{m\times n}(\mathbb{R}) $ consisting of all $ m \times n $ matrices. A basis consists of the matrices $ E_{ij} $ for $ 1 \leq i \leq m $ and $ 1 \leq j \leq n $, where $ E_{ij} $ has a 1 in the $ (i,j) $-th position and 0 elsewhere. Exercise 2 Give a basis for $ V = \mathbb{R}^+ = \{x \mid x \in \mathbb{R} \text{ with } x > 0\} $ with $ x \oplus x' = x \times x' $ and $ c \otimes x = x^c $. ...
Exercise 1 (Multiplication of block matrices). Consider two block matrices $$ A = \begin{pmatrix} A_{11} & \cdots & A_{1t} \\ \vdots & \ddots & \vdots \\ A_{p1} & \cdots & A_{pt} \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} B_{11} & \cdots & B_{1q} \\ \vdots & \ddots & \vdots \\ B_{t1} & \cdots & B_{tq} \end{pmatrix} $$ Moreover, for every $ i \in [p] $, $ j \in [t] $, and $ l \in [q] $ the number of columns of $ A_{ij} $ is equal to the number of rows of $ B_{jl} $. In particular, $ A_{ij} \cdot B_{jl} $ is defined. Prove that $$ A \cdot B = \begin{pmatrix} C_{11} & \cdots & C_{1q} \\ \vdots & \ddots & \vdots \\ C_{p1} & \cdots & C_{pq} \end{pmatrix} $$ with $$ C_{il} = \sum_{j \in [t]} A_{ij} B_{jl} $$ for any $ i \in [p] $ and $ l \in [q] $. ...
Exercise 1 What rows or columns or matrices do you multiply to find the second column of $ AB $ ? the first row of $ AB $ ? the entry in row $ 3 $, column $ 5 $ of $ AB $ ? the entry in row $ 1 $, column $ 1 $ of $ CDE $ ? To find the second column of $ AB $, we multiply matrix $ A $ by the second column of matrix $ B $. ...
Exercise 1 Let $ f:\mathbb{R}\to\mathbb{R} $ be a function. Prove that the following are equivalent: (i) There is a constant $ a\in\mathbb{R} $ such that for every$ x\in\mathbb{R} $ we have$ f(x)=ax $. (ii) For all $ x_1,x_2,c,x\in\mathbb{R} $ we have $ f(x_1+x_2)=f(x_1)+f(x_2) $ and $ f(cx)=c\,f(x) $. (i ⇒ ii) Assume there exists $ a\in\mathbb{R} $ such that $ f(x)=a x $ for all $ x\in\mathbb{R} $ . Then for any $ x_1,x_2,c,x\in\mathbb{R} $ , ...