\[ \newcommand{\parallelsym}{\mathbin{\!/\mkern-5mu/\!}} \newcommand{\tr}{\operatorname{tr}} \] This is a set of problems with which you can take exercise on linear algebra. Day 1. The problems for the first day are related to: Representation of Linear Transformations. Explain why every linear transformation between finite dimensional vector spaces can be regarded as a matrix. (Hint: Consider bases for domain and codomain of the transformation.) Use matrix multiplication to …
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Linear Transformation
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 …
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 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 …