This set of exercises is retrieved from the eighth 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 8.1 Using the recursive definition given in the proof of the existence of determinant, systematically evaluate the determinant of the following matrix: \[A=\begin{pmatrix}1&2&1\\0&1&1\\1&0&2\end{pmatrix}.\] Solution. \[\begin{aligned} \det (A) &= 1 \cdot …
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This set of exercises is retrieved from the seventh 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 7.1 In \(\mathbb{R}^3,\) compute the inner product of \((1,\,2,\,-1)\) and \((2,\,1,\,4).\) What is the length of each vector? What is the angle between these vectors? Solution. The lengths of given …
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. …
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 \(w.\) \[ \begin{pmatrix} 1&2 \\ 0&1 \end{pmatrix} \begin{pmatrix} x&y \\ z&w \end{pmatrix} = \begin{pmatrix} 10&2 \\ 4&2 \end{pmatrix} \] …
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, …
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 …
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 \(G\) …
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 …