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Chapter 1: Q25E (page 1)
Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.
The given coefficient matrix is consistent.
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Consider a dynamical system with two components. The accompanying sketch shows the initial state vector and two eigen vectors of A (with eigen values respectively). For the given values of , draw a rough trajectory. Consider the future and the past of the system.
Determine which of the matrices in Exercises 7鈥�12areorthogonal. If orthogonal, find the inverse.
11. \(\left( {\begin{aligned}{{}}{2/3}&{2/3}&{1/3}\\0&{1/3}&{ - 2/3}\\{5/3}&{ - 4/3}&{ - 2/3}\end{aligned}} \right)\)
Consider a dynamical system with two components. The accompanying sketch shows the initial state vector and two eigenvectors of A (with eigen values respectively). For the given values of , draw a rough trajectory. Consider the future and the past of the system.
Suppose a linear transformation \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) has the property that \(T\left( {\mathop{\rm u}\nolimits} \right) = T\left( {\mathop{\rm v}\nolimits} \right)\) for some pair of distinct vectors u and v in \({\mathbb{R}^n}\). Can Tmap \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\)? Why or why not?
Describe the possible echelon forms of the matrix A. Use the notation of Example 1 in Section 1.2.
a. A is a \({\bf{2}} \times {\bf{3}}\) matrix whose columns span \({\mathbb{R}^{\bf{2}}}\).
b. A is a \({\bf{3}} \times {\bf{3}}\) matrix whose columns span \({\mathbb{R}^{\bf{3}}}\).
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