Linear Algebra

[pgr999]

I have been trying to figure out this math problem for a long time.

Can each vector in R4 be written as a linear combination of the columns of the matrix A below? Why/why not?
matrix A =
[1 2]
[3 4]
[-3 -1]
[6 5]


any help / sugesstions would be appriciated greatly!!!

 

[pka]

There is a very quick answer to the question: no.
R<SUP>4</SUP> must be spanned by 4 independent vectors.
Therefore, no set of fewer that 4 vectors will be a basis.