Let $T,U:V\to W$ be linear transformations. Prove that if $W$ is finite-dimensional, then $\text{rank}(T+U)\leq\text{rank}(T) + \text{rank}(U)$.

$a)$ looks good.

$b)$ follows from $a)$, since $\operatorname{rank}T=\operatorname{dim}R(T)$

(You correctly used the formula for the dimension of the sum of two subspaces.)

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Linear Algebra

Matrix Rank

Solution Verification

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