Question

Find a linearly independent set of vectors that spans the same subspace of R3 as that spanned by the vectors

[-3,1,3] , [-6,5,5],[0,-3,1]

Linearly independent set:
[x,y,z] , [x,y,z]

Answer 1

Here, the subspace spanned by the vectors (-3,1,3), (-6,5,5), (0,-3,1).

Let us consider a relation a(-3,1,3)+b(-6,5,5)+c(0,-3,1) = (,0,0,0) where a, b, c are real numbers.

Then, -3a-6b = 0

a+5b-3c = 0

3a+5b+c = 0

i.e., a = -2b

c = b

Let us take b = k. Then, a = -2k and c = k.

Now we have, -2k(-3,1,3)+k(-6,5,5)+k(0,-3,1) = (,0,0,0)

i.e., (-2)*(-3,1,3)+1*(-6,5,5)+1*(0,-3,1) = (0,0,0)

i.e., 1*(-6,5,5)+1*(0,-3,1) = 2*(-3,1,3)

i.e., (1/2)*(-6,5,5)+(1/2)*(0,-3,1) = (-3,1,3)

Therefore, the first vector (-3,1,3) can be written as the linear combination of other two vectors.

But, the second and third vectors (-6,5,5), (0,-3,1) are linearly independent.

Hence, the required set of linearly independent vectors is = {(-6,5,5),(0,-3,1)}.