Homework 11: One-to-One, Onto

1. Section 8.2, #6, 7, 19(a)(c). Make sure to justify your answer to the question asked in Exercise 7.
2. Determine whether the mappings given in Problems 3 and 4 of Homework 10 are one-to-one. Justify your answers.
3. Let \(T\colon V\to W\) be linear transformation, and let \(\{\bv_1,\ldots,\bv_p\}\) be linearly independent in \(V\).
   1. Prove that if \(T\) is one-to-one, then \(\{T(\bv_1),\ldots,T(\bv_p)\}\) is linearly independent in \(W\).
      Hint. Begin the proof as follows: ``Suppose that \(c_1 T(\bv_1)+\cdots + c_p T(\bv_p)=\bz\). I must show \(c_1=\cdots =c_p=0\).''
   2. Give an example showing that if \(T\) is not one-to-one, then \(\{T(\bv_1),\ldots,T(\bv_p)\}\) need not be linearly independent in \(W\).
4. Let \(T\colon V\to W\) be a linear transformation, with \(\dim V=n\), \(\dim W=m\). Prove the following:
   1. \(\dim (R(T))\leq n\).
   2. \(\dim (R(T))= n\) if and only if \(T\) is one-to-one.
   3. \(\dim (R(T))=m\) if and only if \(T\) is onto.
5. (GRE Math Subject Test) Let \(V\) be the vector space of all \(2\times 3\) matrices, and let \(W\) be the vector space of all \(4\times 1\) column vectors. If \(T\) is a linear transformation from \(V\) onto \(W\), find the dimension of the subspace \(\{ \bv \in V: T(\bv)=\bz \}\). Make sure to justify your answer.