91Ó°ÊÓ

Find the matrix of the given linear transformation \(T\) with respect to the given basis. If no basis is specified, use the standard basis: \(2 \mathrm{x}=\left(1, t, t^{2}\right)\) for \(P_{2}\) $$\mathfrak{A}=\left(\left[\begin{array}{ll}1 & 0 \\\0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\\1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right]\right)$$ for \(\mathbb{R}^{2 \times 2},\) and \(\mathfrak{A}=(1, i)\) for \(\mathbb{C} .\) For the space \(U^{2 \times 2}\) of upper triangular \(2 \times 2\) matrices, use the basis $$\mathfrak{A}=\left(\left[\begin{array}{ll}1 & 0 \\\0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\\0 & 1\end{array}\right]\right)$$ unless another basis is given. In each case, determine whether \(T\) is an isomorphism. If \(T\) isn't an isomorphism, find bases of the kernel and image of \(T,\) and thus deter mine the rank of \(T\). $$\begin{aligned} &T(M)=\left[\begin{array}{ll} 1 & 2 \\ 0 & 3 \end{array}\right] M \text { from } U^{2 \times 2} \text { to } U^{2 \times 2}, \text { with respect }\\\ &\text { to the basis } \mathfrak{B}=\left(\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 1 \\ 0 & 1 \end{array}\right]\right) \end{aligned}$$

Let \(V\) be the space of all infinite sequences of real numbers. See Example \(5 .\) Which of the subsets of \(V\) given in Exercises 12 through 15 are subspaces of \(\boldsymbol{V} ?\) The sequences \(\left(x_{0}, x_{1}, \ldots\right)\) that converge to zero (i.e. \(\lim _{n \rightarrow \infty} x_{n}=0\)

Find the matrix of the given linear transformation \(T\) with respect to the given basis. If no basis is specified, use the standard basis: \(2 \mathrm{x}=\left(1, t, t^{2}\right)\) for \(P_{2}\) $$\mathfrak{A}=\left(\left[\begin{array}{ll}1 & 0 \\\0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\\1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right]\right)$$ for \(\mathbb{R}^{2 \times 2},\) and \(\mathfrak{A}=(1, i)\) for \(\mathbb{C} .\) For the space \(U^{2 \times 2}\) of upper triangular \(2 \times 2\) matrices, use the basis $$\mathfrak{A}=\left(\left[\begin{array}{ll}1 & 0 \\\0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\\0 & 1\end{array}\right]\right)$$ unless another basis is given. In each case, determine whether \(T\) is an isomorphism. If \(T\) isn't an isomorphism, find bases of the kernel and image of \(T,\) and thus deter mine the rank of \(T\). $$T(x+i y)=x-i y \text { from } \mathbb{C} \text { to } \mathbb{C}$$

Find a basis for each of the spaces \(V\) in Exercises 16 through \(36,\) and determine its dimension. The space of all matrices \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\) in \(\mathbb{R}^{2 \times 2}\) such that \(a+d=0\)

Find a basis for each of the spaces \(V\) in Exercises 16 through \(36,\) and determine its dimension. The space of all diagonal \(n \times n\) matrices

Find a basis for each of the spaces \(V\) in Exercises 16 through \(36,\) and determine its dimension. The space of all polynomials \(f(t)\) in \(P_{3}\) such that \(f(1)=0\) and \(\int_{-1}^{1} f(t) d t=0\)

Find the matrix of the given linear transformation \(T\) with respect to the given basis. If no basis is specified, use the standard basis: \(2 \mathrm{x}=\left(1, t, t^{2}\right)\) for \(P_{2}\) $$\mathfrak{A}=\left(\left[\begin{array}{ll}1 & 0 \\\0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\\1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right]\right)$$ for \(\mathbb{R}^{2 \times 2},\) and \(\mathfrak{A}=(1, i)\) for \(\mathbb{C} .\) For the space \(U^{2 \times 2}\) of upper triangular \(2 \times 2\) matrices, use the basis $$\mathfrak{A}=\left(\left[\begin{array}{ll}1 & 0 \\\0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\\0 & 1\end{array}\right]\right)$$ unless another basis is given. In each case, determine whether \(T\) is an isomorphism. If \(T\) isn't an isomorphism, find bases of the kernel and image of \(T,\) and thus deter mine the rank of \(T\). $$T(M)=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right] M-M\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right] \text { from } \mathbb{R}^{2 \times 2} \text { to } \mathbb{R}^{2 \times 2}$$

Find the matrix of the given linear transformation \(T\) with respect to the given basis. If no basis is specified, use the standard basis: \(2 \mathrm{x}=\left(1, t, t^{2}\right)\) for \(P_{2}\) $$\mathfrak{A}=\left(\left[\begin{array}{ll}1 & 0 \\\0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\\1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right]\right)$$ for \(\mathbb{R}^{2 \times 2},\) and \(\mathfrak{A}=(1, i)\) for \(\mathbb{C} .\) For the space \(U^{2 \times 2}\) of upper triangular \(2 \times 2\) matrices, use the basis $$\mathfrak{A}=\left(\left[\begin{array}{ll}1 & 0 \\\0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\\0 & 1\end{array}\right]\right)$$ unless another basis is given. In each case, determine whether \(T\) is an isomorphism. If \(T\) isn't an isomorphism, find bases of the kernel and image of \(T,\) and thus deter mine the rank of \(T\). \(T(M)=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right] M-M\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]\) from \(\mathbb{R}^{2 \times 2}\) to \(\mathbb{R}^{2 \times 2}\) with respect to the basis $$\mathfrak{B}=\left(\left[\begin{array}{rr} 1 & 1 \\ -1 & -1 \end{array}\right],\left[\begin{array}{rr} 1 & -1 \\ 1 & -1 \end{array}\right],\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right],\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]\right)$$

Find the matrix of the given linear transformation \(T\) with respect to the given basis. If no basis is specified, use the standard basis: \(2 \mathrm{x}=\left(1, t, t^{2}\right)\) for \(P_{2}\) $$\mathfrak{A}=\left(\left[\begin{array}{ll}1 & 0 \\\0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\\1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right]\right)$$ for \(\mathbb{R}^{2 \times 2},\) and \(\mathfrak{A}=(1, i)\) for \(\mathbb{C} .\) For the space \(U^{2 \times 2}\) of upper triangular \(2 \times 2\) matrices, use the basis $$\mathfrak{A}=\left(\left[\begin{array}{ll}1 & 0 \\\0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\\0 & 1\end{array}\right]\right)$$ unless another basis is given. In each case, determine whether \(T\) is an isomorphism. If \(T\) isn't an isomorphism, find bases of the kernel and image of \(T,\) and thus deter mine the rank of \(T\). \(T(M)=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right] M-M\left[\begin{array}{ll}1 & 0 \\ 0 & -1\end{array}\right]\) from \(\mathbb{R}^{2 \times 2}\) to \(\mathbb{R}^{2 \times 2},\) with respect to the basis $$\mathfrak{B}=\left(\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right],\left[\begin{array}{rr} 1 & 0 \\ -1 & 0 \end{array}\right],\left[\begin{array}{ll} 0 & 1 \\ 0 & 1 \end{array}\right],\left[\begin{array}{rr} 0 & 1 \\ 0 & -1 \end{array}\right]\right)$$

Find the matrix of the given linear transformation \(T\) with respect to the given basis. If no basis is specified, use the standard basis: \(2 \mathrm{x}=\left(1, t, t^{2}\right)\) for \(P_{2}\) $$\mathfrak{A}=\left(\left[\begin{array}{ll}1 & 0 \\\0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\\1 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right]\right)$$ for \(\mathbb{R}^{2 \times 2},\) and \(\mathfrak{A}=(1, i)\) for \(\mathbb{C} .\) For the space \(U^{2 \times 2}\) of upper triangular \(2 \times 2\) matrices, use the basis $$\mathfrak{A}=\left(\left[\begin{array}{ll}1 & 0 \\\0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}0 & 0 \\\0 & 1\end{array}\right]\right)$$ unless another basis is given. In each case, determine whether \(T\) is an isomorphism. If \(T\) isn't an isomorphism, find bases of the kernel and image of \(T,\) and thus deter mine the rank of \(T\). \(T(M)=\left[\begin{array}{ll}1 & 2 \\ 4 & 3\end{array}\right] M-M\left[\begin{array}{ll}5 & 0 \\ 0 & -1\end{array}\right]\) from \(\mathbb{R}^{2 \times 2}\) to \(\mathbb{R}^{2 \times 2}\)