Consider the following three vectors vector v1 1 3 2 vector

Consider the following three vectors: vector v_1 = [1 -3 2], vector v_2 = [-3 9 -6], vector v_3 = [5 -7 h]. For what value(s) of h is vector v_3 in span {v_1, v_2}? Justify your answer.

Solution

For no value of h is v3 in span{v1,v2}

BEcause note the first two rows of v1 and v2

v1 first two rows has 1 and -3

v2 first two rows has -3 and 9 which is -3 times what v1 has

So any vector in span{v1,v2} will have first two rows as multiple of what v1 has

But v3 has first two rows as 5 and -7 hence not in the span