A linear transformation's image of a basis spans its image

up:: Linear Transformation

Let $T:V\to W$ a linear transformation, and $B\subset V$ a Hamel Basis.

Let $w\in T(V)$. Since it is in $T$‘s image, then there is some vector $v\in V$ such that

$$w=T(v)=T\left(\sum _i{\alpha }_i{v}_i\right)=\sum _i{\alpha }_{i}T(v_i)$$

Thus, $ImT$ is spanned by $\{T(v_i)\}$.

Corollaries