Homework 10: Kernel, Range

Instructions. Assignments should be stapled and written clearly and legibly. Problems 5 and 6 are optional.

1. 8.1, #10, 11, 24.
2. Let \(V\) and \(W\) be vector spaces, and let \(T:V\to W\) be linear. Prove that if \(\{\bv_1,\ldots,\bv_p\}\) spans \(V\), then \(\{T(\bv_1),\ldots,T(\bv_p)\}\) spans \(R(T)\).
3. Define a map \(T:\bR^3 \to F(-\infty,\infty)\) by $$ T\left(\vthree{a}{b}{c}\right) = a \cos^2 x + b \sin^2 x + c $$
   1. Prove that \(T\) is linear.
   2. Find a nonzero vector in \( \ker T\).
4. Define a map \(T:\bR^3 \to F(-\infty,\infty)\) by $$ T\left(\vthree{a}{b}{c}\right) = a \cos x + b \sin x + c $$
   1. Prove that \(T\) is linear.
   2. Find \( \ker T\). Prove your answer.
5. Let \(T:V\to W\) be a linear transformation. Let \(w\) be an element of \(W\), and let \(v_0\) be an element of \(V\) which \(T\) maps to \(w\). Prove that the set of all vectors which \(T\) maps to \(w\) is equal to \(\{v_0+u: u \in \ker T\}\). (Note that this is not, in general, a subspace of \(V\). Why not?)
6. (Differential Equations) Let \(V=W=C^{\infty}(-\infty,\infty)\) be the vector space of infinitely differentiable functions. Let \(T:V\to W\) be the linear transformation $$T(f) = a_mf^{(m)}+a_{m-1}f^{(m-1)}+\cdots + a_1f,$$ where \(f^{(i)}\) denotes the \(i\)-th derivative of \(f\), and \(a_m,a_{m-1},\ldots, a_1\) are constants.
   1. Recall that \(\ker T = \{f\in V: T(f)=0\}\). In terms of differential equations, what does \(\ker T\) represent?
   2. Let \(g\in W\), and let \(H=\{h\in V: T(h)=g\}\). In terms of differential equations, what does \(H\) represent?
   3. Let \(h_0\) be any element of \(H\). Use Problem 5 to prove that \(H=\{h_0+f: f \in \ker T\}\).