Property of Direct Sum

Algebra Level 4

Suppose $V_1$ and $V_2$ are vector spaces over the field F , we have following proposition:

For all vector space $W$ over F and linear maps $f_i:W\rightarrow V_i,i=1,2,$ there exists a unique linear map $\bar f:W\rightarrow V_1\oplus V_2$ such that $f_i=\pi_i\circ\bar f,i=1,2,$ where $\pi_i:V_1\oplus V_2\rightarrow V_i$ denotes canonical projection.

Conversely, if vector space $V$ over F and linear maps $p_i:V\rightarrow V_i,i=1,2$ satisfy following condition:

For all vector space $W$ over F and linear maps $f_i:W\rightarrow V_i,i=1,2,$ there exists a unique linear map $\tilde f:W\rightarrow V$ such that $f_i=p_i\circ\tilde f,i=1,2,$

must $V$ be isomorphic to $V_1\oplus V_2$ ?

Yes No

1 solution

Otto Bretscher
Oct 29, 2018

By the condition stated in the first box, applied to $W=V$ , there exists an $L: V \rightarrow V_1 \oplus V_2$ such that $\pi_i \circ L = p_i$ for $i=1,2$ . Likewise, by the second condition, there exists a $T: V_1 \oplus V_2 \rightarrow V$ such that $p_i \circ T = \pi_i$ . We wish to show that $L$ and $T$ are inverses. By symmetry, it suffices to show that $T\circ L = id_V$ . From the earlier equations we can conclude that $p_i \circ T \circ L = p_i$ . But there exists a unique map $F: V \rightarrow V$ with $p_i \circ F = p_i$ , namely, $F=id_V$ , so that $T\circ L = id_V$ as claimed.

Property of Direct Sum

1 solution

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