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Chapter 1: Q19E (page 39)
Question: Determine whether the statements that follow are true or false, and justify your answer.
19. There exits a matrix A such that.
It is true that there exists a matrix A such that .
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In Exercises 10, write a vector equation that is equivalent tothe given system of equations.
10. \(4{x_1} + {x_2} + 3{x_3} = 9\)
\(\begin{array}{c}{x_1} - 7{x_2} - 2{x_3} = 2\\8{x_1} + 6{x_2} - 5{x_3} = 15\end{array}\)
Determine whether the statements that follow are true or false, and justify your answer.
15: The systemisinconsistent for all matrices A.
Let \(T:{\mathbb{R}^3} \to {\mathbb{R}^3}\) be the linear transformation that reflects each vector through the plane \({x_{\bf{2}}} = 0\). That is, \(T\left( {{x_1},{x_2},{x_3}} \right) = \left( {{x_1}, - {x_2},{x_3}} \right)\). Find the standard matrix of \(T\).
Construct a \(2 \times 3\) matrix \(A\), not in echelon form, such that the solution of \(Ax = 0\) is a plane in \({\mathbb{R}^3}\).
Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be an invertible linear transformation, and let Sand U be functions from \({\mathbb{R}^n}\) into \({\mathbb{R}^n}\) such that \(S\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) and \(\)\(U\left( {T\left( {\mathop{\rm x}\nolimits} \right)} \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\). Show that \(U\left( v \right) = S\left( v \right)\) for all v in \({\mathbb{R}^n}\). This will show that Thas a unique inverse, as asserted in theorem 9. (Hint: Given any v in \({\mathbb{R}^n}\), we can write \({\mathop{\rm v}\nolimits} = T\left( {\mathop{\rm x}\nolimits} \right)\) for some x. Why? Compute \(S\left( {\mathop{\rm v}\nolimits} \right)\) and \(U\left( {\mathop{\rm v}\nolimits} \right)\)).
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