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Homework 6 - Bases and dimension

Due 2021-10-05.

  1. Determine which subsets of the set of vectors \(\begin{pmatrix} -1 \\ 1 \\ 3 \end{pmatrix}, \ \begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix}, \ \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \ \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}\) are linearly independent.

  2. Show that the dimension of the space of solutions to the linear differential equation \(f^{\prime \prime}(x) - f(x) = 0\) is at least two. (Hint: what functions do you know which are equal to their second derivatives?)

  3. Show that \(\dim (V_1 \times V_2) = \dim V_1 + \dim V_2\)

  4. Suppose $U$ is a subspace of a vector space $V$. We define the quotient \(V/U := \lbrace [v] \mid v \in V, [v_1] = [v_2] \text{ if } v_1-v_2 \in U \rbrace\) So for each $v \in V$ we have a symbol $[v]$ and two symbols are equal if the difference of their labels lies in $U$.

    Show that $V/U$ with \([v_1] + [v_2] := [v_1+v_2]\) and \(c[v] := [cv]\) is a vector space. (Note: the key part of this is making sure $+$ and $\cdot$ don’t change if you replace $[v]$ with $[v+u]$ for some $u \in U$.)

  5. Suppose $S$ is a spanning set of $V$ and $U$ is a subspace. Show $[v_i]$ for $ i \in S$ is a spanning set for $V/U$. Conclude that $\dim V/U \leq \dim V$.