Is the following set of vectors a linearly independent subse

Is the following set of vectors a linearly independent subset of M_2, 2? A = {[a_ij] elementof M_2, 2: a_1, 1 + 3a_22 =0}

Solution

No

Because M_{2,2} has dimension 4

SO any subset of M_{2,2} can have atmost 4 linearly independent vectors

And in definition of A there is no restriction on a_{2,1} and a_{1,2}

So A has infinitely many elements. Hence the set is not linearly independent.