00:01
We have a vector space v and a subset of v, s, formed by three vectors, b1, b2, b3, and we know s is linearly independent.
00:14
We have another set or a subset t of the subspace b, and t is formed by three vectors w1, w2, w3, w1 is v1, the same vector here.
00:32
W2 is v1 plus v3 and w3 is v1 plus v2 plus v3.
00:41
So we want to know if t is linearly dependent or independent.
00:50
So to see that we're going to consider any linear combination of the vectors in t equal to the zero vector.
00:59
And if that implies that all coefficients are going to be equal to zero, then t will be independent.
01:07
So let alpha, beta, gamma, b3 real numbers such that the linear combination alpha w1 plus beta w2 plus gamma w3 equals 0.
01:33
Of course the 0 here is the 0 vector of the vector space v.
01:40
So if we have that, you want to know if this implies that the coefficients alpha get beta and gamma got to be all equal to zero.
01:53
Now we replace the definitions of w1, w2 and w3 here given here.
02:03
And so alpha times w1 is v1 plus peta times w2 is b1 plus v3 plus gamma times w3 is v1 plus gama times w3 is v1 plus v2 plus v3...