what's the basis of the subspace of P spanned by
x^2 - 1
x^2 + 1
4x
2x-3
and another basis question...
what's the basis of the space of vectors in a plane with an equation of
2x - 3y + 4z = 0
It will be a subspace of P2. (Polynomials of degree 2). i.e. dim <=3
Check the above vectors for independence.
What is the dim of the null-space? Substitute a few values you will ge the basis.
and another basis question...
what's the basis of the space of vectors in a plane with an equation of
2x - 3y + 4z = 0
As said, just check for linear independence.
In order to do that, use coordinated vectors and express for example $\displaystyle x^2-1=(1,0,-1)_v$ or $\displaystyle 4x=(0,4,0)_v$ and do the same for others and reduce the matrix to the echelon form, if one of the rows turns zero, then that vector makes the set linear dependent, so you just remove it to get your basis.
Write $\displaystyle W=\{(x,y,z)\in\mathbb R^3:2x-3y+4z=0\}$ and make $\displaystyle x,y$ or $\displaystyle z$ (as you like) the subject of that condition, then put it in $\displaystyle (x,y,z)$ and you'll find set wich generates the subspace, so again check for linear independence.