Linear algebra T or F question If its false please provide a

Linear algebra: T or F question. If its false, please provide a counter-example (not an explanation) show that it can be false.

1) Let S1 and S2 be sets of vectors. If Span(S1) = Span (s2), then S1 = S2.

2) A plane in R^3 is a two-dimensional subspace

Solution

1)

False

Consider R2

One basis for R2 is S1={(1,0),(0,1)}

Another basis is S2={(1,1),(1,0)}

S1 and S2 are not equal but

span(S1)=span(S2)=R2

2)

True

A plane is a subspace spanned by two linearly independent vectors