Worksheet 11

1. Determine which subsets of the set \(\begin{pmatrix} 1 \\ -1 \\ 0 \\ -3 \end{pmatrix}, \ \begin{pmatrix} 1 \\ 0 \\ 0 \\ -1 \end{pmatrix}, \ \begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \end{pmatrix}, \ \begin{pmatrix} 3 \\ 2 \\ 3 \\ 2 \end{pmatrix}, \ \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix}\) are linearly dependent and which are linearly independent. (You are going to want to ask Sage for some help.)

2. Find a linearly independent spanning set for the span of the vectors from problem 1.

3. Determine whether the following statements are true or false:

   • Any vector space has a finite linearly independent spanning set.
   • Suppose $V$ is a vector space and $U \subseteq V$ is subspace. If $S$ is a linearly independent set of vectors in $U$, then $S$ is a linearly independent set of vectors in $V$.
   • In the vector space $\operatorname{Func}(\mathbb{R},\mathbb{R})$, the set \(e^x, e^{2x}, e^{3x}, \ldots, e^{mx}\) is linearly independent for any $m > 0$.