Videos for Section 2.5


Section 2.5 is all about building new vector spaces from old ones. There are two kinds of direct sums, namely external and internal.

In an external direct sum, we start with two vector spaces $W_1$ and $W_2$ that may have nothing to do with each other. We then make a new space $V = W_1 \oplus W_2$ consisting of ordered pairs $\begin{pmatrix} {\bf w}_1 \cr {\bf w}_2 \end{pmatrix}$, where ${\bf w}_i \in W_i$. The following video shows how this works, shows how the dimension of $W_1 \oplus W_2$ is the sum of the dimensions of $W_1$ and $W_2$, and shows how to get a basis for $W_1 \oplus W_2$ from bases for $W_1$ and $W_2$.



Sometimes it happens that we have a vector space $V$ and two subspaces $W_1$ and $W_2$. If the dimensions of $W_1$ and $W_2$ add up to the dimension of $V$, and if $W_1 \cap W_2 = \{0\}$, then $V$ is isomorphic to the external direct sum of $W_1$ and $W_2$. In this case we call $V$ the internal direct sum of $W_1$ and $W_2$. We use the same notation as for external direct sum, namely $V = W_1 \oplus W_2$.



There is a third object described in Section 2.5, called the quotient space. We will skip that topic this semester.