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 $N: W \to U$ be a linear transformation.

Then $G \circ N$ is degenerate.

Proof

Recall that the dimension of subspace is not greater than its super space.

Thus the claim follows from:

$\Img {G \circ N} \subseteq \Img G$

$\blacksquare$

Sources

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