Property of Direct Sum 2

Algebra Level pending

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 $g_j:V_j\rightarrow W,j=1,2,$ there exists a unique linear map $\bar g:V_1\oplus V_2\rightarrow W$ such that $g_j=\bar g\circ\iota_j,j=1,2,$ where $\iota_j:V_j\rightarrow V_1\oplus V_2$ denotes canonical injection.

Conversely, if vector space $V$ over F and linear maps $i_j:V_j\rightarrow V,j=1,2$ satisfy following condition:

For all vector space $W$ over F and linear maps $g_j:V_j\rightarrow W,j=1,2,$ there exists a unique linear map $\tilde g:V\rightarrow W$ such that $g_j=\tilde g\circ i_j,j=1,2,$

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

No Yes

1 solution

Brian Moehring
Nov 12, 2018

The following diagram is commutative

so that $\tilde g \circ \tilde f \circ i_j = \tilde g \circ \iota_j = i_j = \text{id}_V \circ i_j \\ \tilde f \circ \tilde g \circ \iota_j = \tilde f \circ i_j = \iota_j = \text{id}_{V_1\oplus V_2} \circ \iota_j$ which, by the uniqueness of the linear maps guaranteed by the propositions, implies $\tilde g \circ \tilde f = \text{id}_V \qquad \tilde f \circ \tilde g = \text{id}_{V_1\oplus V_2} \quad \implies \quad V \simeq V_1\oplus V_2$

Property of Direct Sum 2

1 solution

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