Prove that a system {a1, . . . , an} of vectors is linearly dependent if and only if at least one of the vectors a1, . . . , an can be expressed as a linear combination of the preceding vectors.
mathematics
Description
Question 1 Check whether the system S of vectors a1 = (1, 2, −1), a2 = (−1, 1, 2), a3 = (2, 1, 3) in R 3 is a basis, and if yes, find the coordinate vector of b = (2, 4, 4) in S.
Question 2 Prove that a system {a1, . . . , an} of vectors is linearly dependent
if and only if at least one of the vectors a1, . . . , an can be expressed as a linear
combination of the preceding vectors.
Question 3 Given bases S = {(1, 0, 1),(1, 1, 0),(0, 1, 1)} and S 0 = {(1, 2, 3),(−1, 1, −2),(0, −2, 0)} of R 3 ,
(i) find the transition matrix from S to S 0 , [10]
(ii) compute [x]S if [x]S0 = (1, −1, 1)T
Question 4 Let S and S
0 be two bases of a vector space V and let T be
the transition matrix from S to S
0
. Prove that for every vector x ∈ V ,
[x]S = T · [x]S0.
