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