Determine if the given vectors are basis of the specified ve

Determine if the given vectors are basis of the specified vector spaces: V = M_22 (the vector space of all 2 times 2 matrices). Given vectors: [1 0 1 1] [2 -2 3 2], [1 -1 1 0], [0 -1 1 1], V = M_22. Given vectors: [1 1 1 1], [1 -1 0 0], [0 -1 1 0], [1 0 0 0] V = P_2 (the space of all polynomials of degree lessthanorequalto 2). Given vectors: x^2 + 1, x^2 - 1 and 2x - 1.

Solution

a)

Denote given vectors in the order given as :v1,v2,v3,v4 respectively

v1+v3+v4=v2

Hence the 4 vectors are linearly dependent. A linearly dependent set cannot be basis

Hence,this is not a basis

b)

M22 has the standard basis

e11,e12,e21,e22

where eij is matrix with 1 in i,j entry and all other entries equal to 0

If we can reproduce these standard basis vectors using given vectors then we have proved that they span M22

v4=e11

-(v2-v4)=e12

v3-(v2-v4)=e21

v1-v4+(v2-v4)-(v3-(v2-v4))=e22

c)

(x^2-1+x^2+1)/2=x^2

(x^2+1-(x^2-1))/2=1

(2x-1 +(x^2+1-(x^2-1))/2)/2=x

HEnce, given vectors span P2 as their span contains the standard basis for P2