91Ó°ÊÓ

Label the following statements as true or false. In each part, \(V\) and \(W\) are finite-dimensional vector spaces (over \(F\) ), and \(\mathrm{T}\) is a function from \(\mathrm{V}\) to \(\mathrm{W}\). (a) If \(\mathrm{T}\) is linear, then \(\mathrm{T}\) preserves sums and scalar products. (b) If \(\mathrm{T}(x+y)=\mathrm{T}(x)+\mathrm{T}(y)\), then \(\mathrm{T}\) is linear. (c) \(\mathrm{T}\) is one-to-one if and only if the only vector \(x\) such that \(\mathrm{T}(x)=0\) is \(x=0\). (d) If \(T\) is linear, then \(T\left(0_{v}\right)=0_{w}\). (e) If \(T\) is linear, then nullity \((T)+\operatorname{rank}(T)=\operatorname{dim}(W)\). (f) If \(\mathrm{T}\) is linear, then \(\mathrm{T}\) carries linearly independent subsets of \(\mathrm{V}\) onto linearly independent subsets of W. (g) If \(\mathrm{T}, \mathrm{U}: \mathrm{V} \rightarrow \mathrm{W}\) are both linear and agree on a basis for \(\mathrm{V}\), then \(T=U\). (h) Given \(x_{1}, x_{2} \in \mathrm{V}\) and \(y_{1}, y_{2} \in \mathrm{W}\), there exists a linear transformation \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}\) such that \(\mathrm{T}\left(x_{1}\right)=y_{1}\) and \(\mathrm{T}\left(x_{2}\right)=y_{2}\). For Exercises 2 through 6, prove that \(\mathrm{T}\) is a linear transformation, and find bases for both \(N(T)\) and \(R(T)\). Then compute the nullity and rank of \(T\), and verify the dimension theorem. Finally, use the appropriate theorems in this section to determine whether \(\mathrm{T}\) is one-to-one or onto.

For each of the following linear transformations \(\mathrm{T}\), determine whether \(T\) is invertible and justify your answer. (a) \(\mathrm{T}: \mathrm{R}^{2} \rightarrow \mathrm{R}^{3}\) defined by \(\mathrm{T}\left(a_{1}, a_{2}\right)=\left(a_{1}-2 a_{2}, a_{2}, 3 a_{1}+4 a_{2}\right)\). (b) \(\mathrm{T}: \mathrm{R}^{2} \rightarrow \mathrm{R}^{3}\) defined by \(\mathrm{T}\left(a_{1}, a_{2}\right)=\left(3 a_{1}-a_{2}, a_{2}, 4 a_{1}\right)\). (c) \(\mathrm{T}: \mathrm{R}^{3} \rightarrow \mathrm{R}^{3}\) defined by \(\mathrm{T}\left(a_{1}, a_{2}, a_{3}\right)=\left(3 a_{1}-2 a_{3}, a_{2}, 3 a_{1}+4 a_{2}\right)\). (d) \(\mathrm{T}: \mathrm{P}_{3}(R) \rightarrow \mathrm{P}_{2}(R)\) defined by \(\mathrm{T}(p(x))=p^{\prime}(x)\). (e) \(\mathrm{T}: \mathrm{M}_{2 \times 2}(R) \rightarrow \mathrm{P}_{2}(R)\) defined by \(\mathrm{T}\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)=a+2 b x+(c+d) x^{2} .\) (f) \(\mathrm{T}: \mathrm{M}_{2 \times 2}(R) \rightarrow \mathrm{M}_{2 \times 2}(R)\) defined by \(\mathrm{T}\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)=\left(\begin{array}{cc}a+b & a \\ c & c+d\end{array}\right)\).

Let \(V=R^{3}\), and define \(f_{1}, f_{2}, f_{3} \in V^{*}\) as follows: $$ \mathrm{f}_{1}(x, y, z)=x-2 y, \quad \mathrm{f}_{2}(x, y, z)=x+y+z, \quad \mathrm{f}_{3}(x, y, z)=y-3 z . $$ Prove that \(\left\\{\mathrm{f}_{1}, \mathrm{f}_{2}, \mathrm{f}_{3}\right\\}\) is a basis for \(\mathrm{V}^{*}\), and then find a basis for \(\mathrm{V}\) for which it is the dual basis.

Let \(T\) be the linear operator on \(R^{2}\) defined by $$ \mathrm{T}\left(\begin{array}{l} a \\ b \end{array}\right)=\left(\begin{array}{l} 2 a+b \\ a-3 b \end{array}\right) $$ let \(\beta\) be the standard ordered basis for \(\mathrm{R}^{2}\), and let $$ \beta^{\prime}=\left\\{\left(\begin{array}{l} 1 \\ 1 \end{array}\right),\left(\begin{array}{l} 1 \\ 2 \end{array}\right)\right\\} . $$ Use Theorem \(2.23\) and the fact that $$ \left(\begin{array}{ll} 1 & 1 \\ 1 & 2 \end{array}\right)^{-1}=\left(\begin{array}{rr} 2 & -1 \\ -1 & 1 \end{array}\right) $$ to find \([\mathrm{T}]_{\beta^{\prime}}\).

In \(\mathrm{R}^{2}\), let \(L\) be the line \(y=m x\), where \(m \neq 0\). Find an expression for \(\mathrm{T}(x, y)\), where (a) \(\mathrm{T}\) is the reflection of \(\mathrm{R}^{2}\) about \(L\). (b) \(\mathrm{T}\) is the projection on \(L\) along the line perpendicular to \(L\). (See the definition of projection in the exercises of Section 2.1.)

Let \(\mathrm{V}\) be an \(n\)-dimensional vector space with an ordered basis \(\beta\). Define \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{F}^{n}\) by \(\mathrm{T}(x)=[x]_{\beta}\). Prove that \(\mathrm{T}\) is linear.

Prove that a function T : F11 ~ F'11 is linear if and only if there exist h , f2, .. . , fm E (Fn)• such that T(x) = (h (x), f2(x), .. . , fm(x)) for all x E F11 • Hint: If Tis linear, define fi(x) = (gi T)(x) for x E Fn; that is, fi = Tt(gi) for 1 :5 i :5 m, where {g1 , g2, .. . , gm} is the dual basis of the standard ordered basis for Fm.

Find linear transformations U, T: F \(^{2} \rightarrow \mathrm{F}^{2}\) such that UT \(=\mathrm{T}_{0}\) (the zero transformation) but \(\mathrm{TU} \neq \mathrm{T}_{0}\). Use your answer to find matrices \(A\) and \(B\) such that \(A B=O\) but \(B A \neq O\).

Let \(\mathrm{V}\) be the vector space of complex numbers over the field \(R\). Define \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{V}\) by \(\mathrm{T}(z)=\bar{z}\), where \(\bar{z}\) is the complex conjugate of \(z\). Prove that \(\mathrm{T}\) is linear, and compute \([\mathrm{T}]_{\beta}\), where \(\beta=\\{1, i\\}\). (Recall by Exercise 39 of Section \(2.1\) that \(T\) is not linear if \(V\) is regarded as a vector space over the field \(C\).)

Let \(\mathrm{V}\) be an \(n\)-dimensional vector space, and let \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{V}\) be a linear transformation. Suppose that \(\mathrm{W}\) is a T-invariant subspace of \(\mathrm{V}\) (see the exercises of Section 2.1) having dimension \(k\). Show that there is a basis \(\beta\) for \(\mathrm{V}\) such that \([\mathrm{T}]_{\beta}\) has the form $$ \left(\begin{array}{ll} A & B \\ O & C \end{array}\right), $$ where \(A\) is a \(k \times k\) matrix and \(O\) is the \((n-k) \times k\) zero matrix.