Sum of Degenerate Linear Transformation is Degenerate

Theorem

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

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

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

Then $S + T$ is a degenerate linear transformation.

Proof

Let $\set {s_1, \ldots, s_m}$ be a generator of $\Img S$.

Let $\set {t_1, \ldots, t_n}$ be a generator of $\Img T$.

Then $\set {s_1, \ldots, s_m, t_1, \ldots, t_n}$ is a generator of $\Img {S + T}$.

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

$\map \dim {\Img {S + T}} \le m + n$

$\blacksquare$

Sources

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