This set of exercises is retrieved from the sixth chapter of Linear Algebra by Robert J. Valenza. Note that these solutions are not fully elaborated; You have to fill the descriptions by yourself. Problem 6.1 Let \(T:\mathbb{R}^2 \rightarrow \mathbb{R}\) be a linear transformation and suppose that \(T(1,\,1)=5\) and \(T(0,\,1)=2.\) Find \(T(x_1,\,x_2)\) for all \(x_1,\) \(x_2 \in \mathbb{R}.\) Solution. Suppose \((x_1 ,\,x_2 )\) be given. Take \(\lambda_1 = x_1 ,\) \(\lambda_2 = x_2 – x_1 ,\) then \[(x_1 ,\,x_2 ) = …
This set of exercises is retrieved from the fifth chapter of Linear Algebra by Robert J. Valenza. Note that these solutions are not fully elaborated; You have to fill the descriptions by yourself. Problem 5.1 Solve the following matrix equation for \(x,\) \(y,\) \(z\) and \(w.\) \[ \begin{pmatrix} 1&2 \\ 0&1 \end{pmatrix} \begin{pmatrix} x&y \\ z&w \end{pmatrix} = \begin{pmatrix} 10&2 \\ 4&2 \end{pmatrix} \] Solution. Taking \(R_1 \,\leftarrow\, R_1 – 2R_2 ,\) we obtain \[\left( \begin{array}{cc|cc} 1 & 2 & …
This set of exercises is retrieved from the fourth chapter of Linear Algebra by Robert J. Valenza. Note that these solutions are not fully elaborated; You have to fill the descriptions by yourself. Problem 4.1 Let \(v_1 ,\) \(\cdots ,\) \(v_n \) be linearly independent family in a vector space \(V.\) Show that if \(i\ne j,\) then \(v_i \ne v_j .\) In other words, a linearly independent family can never contain a repeated vector. Solution. Suppose not, that is, suppose …
This set of exercises is retrieved from the third chapter of Linear Algebra by Robert J. Valenza. Note that these solutions are not fully elaborated; You have to fill the descriptions by yourself. Problem 3.1 Show that the solution set \(W\) of vectors \((x_1 ,\,x_2 )\) in \(\mathbb{R}^2\) satisfying the equation \[x_1 + 8x_2 = 0\] is a subspace of \(\mathbb{R}^2 .\) Solution. The solution set is \[S = \left\{ (-8s,\, s) \,\vert\, s\in\mathbb{R} \right\}.\] This set is closed under …
This set of exercises is retrieved from the second chapter of Linear Algebra by Robert J. Valenza. Note that these solutions are not fully elaborated; You have to fill the descriptions by yourself. Problem 2.1 Give an example of a noncommutative group of \(24\) elements. Solution. \(S_4 .\) Problem 2.2 Give an example of a group \(G\) and a nonempty subset \(H\) of \(G\) which is closed under the operation defined on \(G,\) but is not a subgroup of \(G.\) …
This set of exercises is retrieved from the second chapter of Linear Algebra by Robert J. Valenza. Note that these solutions are not fully elaborated; You have to fill the descriptions by yourself. Problem 1.1 Find the sets \(S,\) \(T\) and \(U\) and functions \(f: S \rightarrow T\) and \(g: T \rightarrow U\) such that \(g \circ f\) is injective, but \(g\) is not injective. Solution. Take \(S = U = \left\{ 1 \right\} ,\,\, T = \left\{ 0,\,1 \right\},\) …