Sum of Degenerate Linear Transformation is Degenerate
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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