91Ó°ÊÓ

Give a geometric description of the linear transformation defined by the elementary matrix. $$A=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]$$

Give a geometric description of the linear transformation defined by the matrix product. $$A=\left[\begin{array}{ll} 0 & 3 \\ 1 & 0 \end{array}\right]=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]\left[\begin{array}{ll} 1 & 0 \\ 0 & 3 \end{array}\right]$$

True or False? In Exercises 57 and \(58,\) determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) A linear transformation is operation preserving if the same result occurs whether the operations of addition and scalar multiplication are performed before or after the linear transformation is applied. (b) The function \(g(x)=x^{3}\) is a linear transformation from \(R\) into \(R\) (c) Any linear function of the form \(f(x)=a x+b\) is a linear transformation from \(R\) into \(R\)

Use the concept of a fixed point of a linear transformation \(T: V \rightarrow V .\) A vector \(\mathbf{u}\) is a fixed point if \(T(\mathbf{u})=\mathbf{u}\) (a) Prove that 0 is a fixed point of any linear transformation \(T: V \rightarrow V\) (b) Prove that the set of fixed points of a linear transformation \(T: V \rightarrow V\) is a subspace of \(V\) (c) Determine all fixed points of the linear transformation \(T: R^{2} \rightarrow R^{2}\) represented by \(T(x, y)=(x, 2 y)\) (d) Determine all fixed points of the linear transformation \(T: R^{2} \rightarrow R^{2}\) represented by \(T(x, y)=(y, x)\)

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The composition \(T\) of linear transformations \(T_{1}\) and \(T_{2}\) represented by \(T(\mathbf{v})=T_{2}\left(T_{1}(\mathbf{v})\right),\) is defined if the range of \(T_{1}\) lies within the domain of \(T_{2}\) (b) In general, the compositions \(T_{2} \circ T_{1}\) and \(T_{1} \circ T_{2}\) have the same standard matrix \(A\) (c) If \(T: R^{n} \rightarrow R^{n}\) is an invertible linear transformation with standard matrix \(A,\) then \(T^{-1}\) has the same standard matrix \(A\)

Let \(T: M_{2,3} \rightarrow M_{3,2}\) be represented by \(T(A)=A^{T}\). Find the matrix for \(T\) relative to the standard bases for \(M_{2,3}\) and \(M_{3,2}\).

Let \(T_{1}: V \rightarrow V\) and \(T_{2}: V \rightarrow V\) be one-to-one linear transformations. Prove that the composition \(T=T_{2} \circ T_{1}\) is one-to- one and that \(T^{-1}\) exists and is equal to \(T_{1}^{-1} \circ T_{2}^{-1}\). Getting Started: To show that \(T\) is one-to-one, you can use the definition of a one-to-one transformation and show that \(T(\mathbf{u})=T(\mathbf{v})\) implies \(\mathbf{u}=\mathbf{v} .\) For the second statement, you first need to use Theorems 6.8 and 6.12 to show that \(T\) is invertible, and then show that \(T \circ\left(T_{1}^{-1} \circ T_{2}^{-1}\right)\) and \(\left(T_{1}^{-1} \circ T_{2}^{-1}\right) \cdot T\) are identity transformations. (i) Let \(T(\mathbf{u})=T(\mathbf{v}) .\) Recall that \(\left(T_{2} \circ T_{1}\right)(\mathbf{v})=T_{2}\left(T_{1}(\mathbf{v})\right)\) for all vectors \(\mathbf{v} .\) Now use the fact that \(T_{2}\) and \(T_{1}\) are one-to-one to conclude that \(\mathbf{u}=\mathbf{v}\) (ii) Use Theorems 6.8 and 6.12 to show that \(T_{1}, T_{2},\) and \(T\) are all invertible transformations. So \(T_{1}^{-1}\) and \(T_{2}^{-1}\) exist. (iii) Form the composition \(T^{\prime}=T_{1}^{-1} \circ T_{2}^{-1} .\) It is a linear transformation from \(V\) to \(V .\) To show that it is the inverse of \(T,\) you need to determine whether the composition of \(T\) with \(T^{\prime}\) on both sides gives an identity transformation.