Proof
The mapping
has the property
-
![{\displaystyle {}\varphi (v_{j})=\sum _{i=1}^{m}s_{ij}w_{i}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/915cb5c634553af51dcd859dc3173a33230ec7c7)
where
is the
-th entry of the
-th column vector. Therefore,
-
![{\displaystyle {}\varphi {\left(\sum _{j=1}^{n}a_{j}v_{j}\right)}=\sum _{j=1}^{n}a_{j}{\left(\sum _{i=1}^{m}s_{ij}w_{i}\right)}=\sum _{i=1}^{m}{\left(\sum _{j=1}^{n}a_{j}s_{ij}\right)}w_{i}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be3e38d12cce2b33396bf01dd984ab9af05fb54b)
This is
if and only if
for all
, and this is equivalent with
-
![{\displaystyle {}\sum _{j=1}^{n}a_{j}s_{j}=0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7618bf456894b6470d7fa30805c4d2df1fff685)
For this vector equation, there exists a nontrivial tuple
, if and only if the columns are linearly dependent, and this holds if and only if
is not injective.