Right Product with Degenerate Linear Transformation is Degenerate

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Theorem

Let $U, V, W$ be vector spaces over a field $K$.

Let $G: U \to V$ be a degenerate linear transformation.

Let $M: V \to W$ be a linear transformation.

Then $M \circ G$ is degenerate.

Proof

Let $\set {s_1, \ldots, s_n}$ be a generator of $\Img G$.

Then $\set {\map M {s_1}, \ldots, \map M {s_n} }$ is a generator of $\Img {M \circ G}$.

By Cardinality of Generator of Vector Space is not Less than Dimension:

$\map \dim {\Img {M \circ G}} \le n$

$\blacksquare$

Sources

• 2002: Peter D. Lax: Functional Analysis: $2.2$: Index of a Linear Map