form a basis for R3 if and only if k=/

Question

The vectors:

v1=[6 -1 0], v2=[-5 3 7], v3=[2 4 k]

form a basis for R³ if and only if k=/

Answer 1

We want to put this in to our matrix. since, we want to have R³ then we need to have our 3 vectors span that region. And so we need to solve for k so that all three vectors are Linearly Independent.

so,

A = [ 6 -5 2 ]

[ -1 3 4 ]

[ 0 7 k ]

step 1: sove for det (a) do the cross product to solve for each of the smaller squares.

det(A) = 6 * (3k-28) -(-5)(-k) + 2(-7)

= 18k - 168 - 5k -14

= 13k-182

step 2: solve for k so

13k = 182

k = 14

V1, V2, V3 are Linearly Independent if Det(a) does not equal k=14.

Thus, we can say that V1, V2, V3 form a basis for R³ iff. k does not equal 14.