Linear Algebra Find a subset of vectors that forms a basis f

Linear Algebra

Find a subset of vectors that forms a basis for the space spanned by v1 = (1,-1,5,2), v2 = (2,-3,-1,0), v3 = (4,-5,9,4), v4 = (0,4,2,-3), v5 = (3,-8,2,5). Express the other vector(s) as a linear combination of the basis vectors.

Solution

V1 = 1, -1, 5, 2
V2 = 2, -3, -1, 0
V3 = 4, -5, 9, 4
V4 = 0, 4, 2, -3
V5 = 3, -8, 2, 5

Consider the first 4 (V1 - V4) vectors, following vectors can be formed

V6 = 2V1 - V2 = 0, 1, 9, 4
V7 = 4V1 - V3 = 0, 1, 11, 6
V8 = V7 - V6   = 0, 0, 2, 2
V9 = 4V6 - V4 = 0, 0, 34, 17
V10 = 17V8 - V9 = 0, 0 0, 17
ie
V1 = 1, -1, 5, 2
V6 = 0, 1, 9, 4
V9 = 0, 0, 34, 17
V10 = 0, 0 0, 17

all these above vecotrs are linearly independant
so first four vectos can form the basis for the subspace.

now we can write
aV1 + bV2 + cV3 + dV4 = V5
ie,
a + 2b + 4c = 3
-a - b - 5c + 4d= -8
5a -b +9c +2d = 2
2a + 4c - 3d= 5

solving these equations

a + 2b + 4c = 3
b - c + 4d= -5
11c - 23d =34
8c -13d= 21
a = -1, b = 0, c = 1, d = -1
ie,vector V5 = V3 - V4 - V1