Which of the sets that follow are spanning sets for P3 Justi

Which of the sets that follow are spanning sets for P_3 Justify your answers. {1, x^2, x^2 - 2} {2,x^2,x,2x + 3} {x + 2,x + 1,x^2 - 1} {x + 2,x^2 - 1}

Solution

(a)

Denote vectors by:

v1=1,v2=x^2,v3=x^2-2

v2-2v1=v3

Hence this is a linearly dependent set but P3 has dimension 3 so this is not a spanning set for P3

(b)

2/2=1

Hence standard basis vectors for P3: 1,x,x^2 belong to the span of this set. Hence this is a spanning set.

(c)

Denote: v1=x+2,v2=x+1,v2=x^2-1

v1-v2=1

v2-(v1-v2)=x+1-1=x

v3+(v1-v2)=x^2-1+1=x^2

So all standard basis vectors of P3 are in the span of this set. Hence this is a spanning set.

(d)

The set has only 3 vectors but P3 has dimension 3 so it cannot be a spanning set.