Help with linear mapping!

[n_eko]

Let L : C³ -->M_2x2(C) be a linear map. Prove that there exists a vector v E M_2x2(C) such that v is not in Range(L).

I'm not sure where to start, any help would be appreciated!

 

[HallsofIvy]

\(\displaystyle C^3\), the set of ordered triples of complex numbers, has dimension 3 as a vector space over the complex numbers. \(\displaystyle M_{2\times 2}\), the set of 2 by 2 matrices with complex entries, has dimension 4. A range (image) of any linear transformation from \(\displaystyle C^3\) can have dimension at most 3.

 

[Steven G]

Let L : C³ -->M_2x2(C) be a linear map. Prove that there exists a vector v E M_2x2(C) such that v is not in Range(L).

I'm not sure where to start, any help would be appreciated!

One place to start is making sure that you understand the mapping. Can you show us an example of a mapping from C³ to M_2x2(C) under the mapping L?