The document discusses vectors v1, v2, v3, and v4 and the subspaces W and V. It determines that: (1) v1 and v2 are linearly independent vectors that span the subspace W; (2) V has dimension 3 and is spanned by v1, v2, and v4; (3) v4 is linearly independent of v1 and v2 and outside of the subspace W.
1. Outside of circle is 2v4+3v3
Solution
from the figure of circle denoting W, it is easily see v1 and v2 are linearly
independent. there is a linear combination v2 + 3v3 in the circle. that means v3 is also a vector of
W. outside W, 2v4+ v2 is given. that means v2 and v4 are the vectors outside W. from this
observation, we follow (1) {v1,v2} are linearly independent is true. (2) V is spanned by v1,v2 ,
v4 only. so, dimension of V = 3 (3) W is spanned by v1, v2 is true. (4) 2v4 + v2 is not in W
means v4 is linearly independent of V2. thus, W is spanned by v1 and v2. outside W is spanned
by v4.