Linear algebra truefalse questions A subset of P3R containin

A subset of P_3(R) containing 4 functions is linearly dependent. Every subset of P_3(R) containing 5 functions is linearly dependent. There is a 5 function subset of P_3(R) that spans all of P_3(R). There is a 2 vector subset of R^3 that spans all of R^3. There is a 4 vector subset of R^3 that spans all of R^3. A subset of R^4 that is linearly independent and spans all of R^4 contains exactly 5 vectors

Solution

1. False; For example {1,x,x^2,x^3} is subset of four function which is linearly independent.In fact this subset forms a basis for P_3(R).

2. True. Since dimension of P_3(R) is four. Any subset containing 5 vectors must be linearly dependent; otherwise P_3(R) would have dimension greater or equal to 5 which is a contradiction.

3.True. For example Take S={1,x,x^2,x^3,1+x}. Since {1,x,x^2,x^3} is subset of S, span of S must contain span of {1,x,x^2,x^3}. Cosequently span of S is equal to all of P_3(R).

4.False. Otherwise dimension of R^3 would have been less or equal to 2 but that\'s a contradiction since we know dimension of R^3 is equal to 3.

5.True. For example take S ={(1,0,0),(0,1,0),(0,0,1),(1,1,1)}. Since S contains a basis one must have span S equal to R^3.

6. False. For example take S= {(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)}. Span of S is all of R^4 and S contains exactly four element.