(a) Show that B {[2, 3, 0,1],[1, 1, 1,1]} is a maximal linearly independent subset of...

(a) Show that B {[2, 3, 0,1],[1, 1, 1,1]} is a maximal linearly independent subset of S {[1, 4, 1,2],[1, 1, 1,1],[3, 2,1, 0],[2, 3, 0,1]}.

(b) Calculate dim(span(S)).

Solutions

Expert Solution

(a). A set of vectors is maximally linearly independent if including any other vector in the vector space would make it linearly dependent.

It may be observed that the vectors (2,3,1,0) and (1,1,1,1) are common to the sets B and S.

Let A =

2	1	1	3
3	1	4	2
0	1	1	1
1	1	2	0

The RRF of A is

1	0	0	0
0	1	0	0
0	0	1	0
0	0	0	1

It implies that the set B would remain linearly independent even after including the 2 vectors from the set S which are not in B. Therefore, the set B is NOT a maximal linearly independent subset of the set S. ( there seems to a misprint in the question).

(b). It is apparent from the RREF of A that the 4 vectors in S are linearly independent so that dim(S) = 4.

(c ). Since dim (R⁴ ) = 4 which is same as dim(S), and since the set S is linearly independent hence S is also a basis for R⁴ . Hence S spans R⁴ .

milcah answered 1 year ago

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