Question

Use the solution method from this example to find a basis for the given subspace. 1 4 0 5 1 S = span -1 0 -1 4 0 5 Give the dimension of the basis.

Answer 1

Given 15 FINANS S = span 4 It is sufficient to find the linearly independent vectors, By definition, we say that the vectors (V1, V2,..., Vx} is linearly dependent if there exists scalars not all of the zerro such that cv + a2y +...+char = 7 By definition, we say that the vectors (V1, V2,..., Ve} is linearly independent lif V + V +...+ V: = 7 then c = 3 = ... = (n = 0. 5 4 1 consider a + b +c +d -1 - 1 0 5 1 1 5 1 1 5 1 - 1 0 1 1 1 -1 1 5 0 1 40 4 0 0 b 0 Let A= -1 0 4 -1 0 4 1 4 0 5 | 0 4 0 Applying Rs → R+R, R → R - R,we get 1 4 0 5 0 1 1 || 0 4 4 4 0 0 0 0 Applying Rg → R3 -4Ry, we get [i 4 O 5110 0 1 1 1 b ,isin echelon form. 0 0 0 0 0 0 0 0 0 2010 2 d Basis for S = 08 in echelon form, first, second columns have pivot elements Dim(S) = 2