91Ó°ÊÓ

Let \(A\) and \(B\) be similar matrices and let \(\lambda\) be any scalar. Show that (a) \(A-\lambda I\) and \(B-\lambda I\) are similar (b) \(\operatorname{det}(A-\lambda I)=\operatorname{det}(B-\lambda I)\)

Let \(L\) be a linear operator on \(\mathbb{R}^{n} .\) Suppose that \(L(\mathbf{x})=\mathbf{0}\) for some \(\mathbf{x} \neq \mathbf{0} .\) Let \(A\) be the matrix representing \(L\) with respect to the standard basis \(\left\\{\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{n}\right\\} .\) Show that \(A\) is singular.

Determine the kernel and range of each of the following linear operators on \(\mathbb{R}^{3}:\) (a) \(L(\mathbf{x})=\left(x_{3}, x_{2}, x_{1}\right)^{T}\) (b) \(L(\mathbf{x})=\left(x_{1}, x_{2}, 0\right)^{T}\) (c) \(L(\mathbf{x})=\left(x_{1}, x_{1}, x_{1}\right)^{T}\)

Find the kernel and range of each of the following linear operators on \(P_{3}\) (a) \(L(p(x))=x p^{\prime}(x)\) (b) \(L(p(x))=p(x)-p^{\prime}(x)\) (c) \(L(p(x))=p(0) x+p(1)\)

Let \(V\) and \(W\) be vector spaces with ordered bases \(E\) and \(F,\) respectively. If \(L: V \rightarrow W\) is a linear transformation and \(A\) is the matrix representing \(L\) relative to \(E\) and \(F,\) show that (a) \(\mathbf{v} \in \operatorname{ker}(L)\) if and only if \([\mathbf{v}]_{E} \in N(A)\) (b) \(\mathbf{w} \in L(V)\) if and only if \([\mathbf{w}]_{F}\) is in the column space of \(A\)