Learning Guide Learning Guide

thenc 1 = c 2 = c 3 = · · · = c n = 0.3.8 The Maple Packages • 89Problem: Determine a basis for the vector space generated by the vectors[1, −1, 0, 1], [5, −2, 3, −1], and [6, −3, 3, 0]. Express the vector [1, 2, 3, −5]with respect to this basis.Solution: Enter the vectors.> with(LinearAlgebra):> v1:=:> v2:=:> v3:=:> vector_space:=;⎡⎤1 −1 0 1vector_space := ⎣ 5 −2 3 −1 ⎦6 −3 3 0If the vectors are linearly independent, then they form a basis. To testlinear independence, set up the equation c 1 v 1 + c 2 v 2 + c 3 v 3 = 0c 1 [1, −1, 0, 1] + c 2 [5, −2, 3, −1] + c 3 [6, −3, 3, 0] = [0, 0, 0, 0]which is equivalent toc 1 + 5c 2 + 6c 3 = 0−c 1 − 2c 2 − 3c 3 = 03c 2 + 3c 3 = 0c 1 − c 2 = 0> LinearSolve( Transpose(vector_space), );⎡ ⎤−_t0 3⎣ −_t0 3⎦_t0 3The vectors are linearly dependent since each is a linear product of avariable. Thus, they cannot form a basis. The RowSpace command returnsa basis for the vector space.> b:=RowSpace(vector_space);