Homework 9: Linear Transformations

Instructions. Assignments should be stapled and written clearly and legibly.

1. Let \(T:V\to W\) be a linear transformation, and let \(\{\bv_1,\bv_2,\ldots,\bv_n\}\) be a set of vectors that spans \(V\). Prove that if $$ T(\bv_1)=T(\bv_2)=\cdots=T(\bv_n)=\mathbf{0}, $$ then \(T(\bu)=\mathbf{0}\) for all vectors \(\bu\) in \(V\).
   Hint. Use Theorem 2.1.
2. Let \(T:V\to W\) be a linear transformation.
   1. Suppose that \(\{\bv_1,\ldots,\bv_p\}\) is a linearly dependent set of vectors in \(V\). Using the definition of linear dependence and Theorem 2.1, prove that \(\{T(\bv_1),\ldots,T(\bv_p)\}\) is linearly dependent in \(W\).
      Hint. Begin the proof as follows: ``Since \(\{\bv_1,\ldots,\bv_p\}\) is linearly dependent, there exist scalars \(c_1,\ldots,c_p\) which are not all zero for which \(c_1\bv_1+\cdots+c_p\bv_p=\mathbf{0}\).''
   2. Suppose that \(\{\bv_1,\ldots,\bv_p\}\) is a linearly independent set of vectors in \(V\). Is \(\{T(\bv_1),\ldots,T(\bv_p)\}\) necessarily linearly independent in \(W\)? Is yes, prove it. If no, give a counterexample.
3. Determine whether each of the following transformations \(T\) is linear. Prove your answers.
   1. \(T:\bR^2\to \bR^2\) defined by \(T(\vtwo{x_1}{x_2})=\vtwo{4x_1-2x_2}{3|x_2|}\).
   2. \(T:\bR^2\to \bR^3\) defined by \(T(\vtwo{x_1}{x_2})=\vthree{x_1-2x_2}{x_1+3x_2}{3x_1-4x_2}\).
   3. \(T:P_2\to P_2\) defined by \(T(a+bx+cx^2)=a + b(x+1)+b(x+1)^2\).
4. Show that each of the following transformations \(T:\bR^2\to \bR^2\) is linear by finding a matrix \(A\) such that \(T(\bx)=A\bx\). Describe geometrically what each transformation does.
   1. \(T(\vtwo{x_1}{x_2})=\vtwo{-x_1}{\pe x_2}\)
   2. \(T(\vtwo{x_1}{x_2})=\vtwo{x_2}{x_1}\)
   3. \(T(\vtwo{x_1}{x_2})=\vtwo{0}{x_2}\)
   4. \(T(\bx)=-\bx\)
   5. \(T(\bx)=\frac{1}{2}\bx\)