# Subspace of P3, linearly independence?

## Homework Statement

Let U be the subspace of P3(ℝ) spanned by
E={x^3,x^3-x^2,x^3+x^2,x^3-1}
find a linearly independent subset F of E spanning U.

## Homework Equations

E={x^3,x^3-x^2,x^3+x^2,x^3-1}

## The Attempt at a Solution

a(x^3)+b(x^3-x^2)+c(x^3+x^2)+d(x^3-1)=0x^3+0x^2+0x+0
(a+b+c+d)x^3+(-b+c)x^2+(-d)=0x^3+0x^2+0x+0

a+b+c+d=0
-b+c=0
-d=0

a=-2b, b=c, d=0
t(-2,1,1,0), t$\in$ℝ

How do I figure out which vectors are linearly independent??

## Homework Statement

Let U be the subspace of P3(ℝ) spanned by
E={x^3,x^3-x^2,x^3+x^2,x^3-1}
find a linearly independent subset F of E spanning U.

## Homework Equations

E={x^3,x^3-x^2,x^3+x^2,x^3-1}

## The Attempt at a Solution

a(x^3)+b(x^3-x^2)+c(x^3+x^2)+d(x^3-1)=0x^3+0x^2+0x+0
(a+b+c+d)x^3+(-b+c)x^2+(-d)=0x^3+0x^2+0x+0

a+b+c+d=0
-b+c=0
-d=0

a=-2b, b=c, d=0
t(-2,1,1,0), t$\in$ℝ

How do I figure out which vectors are linearly independent??

Just say that P^3 is isomorphic to R^4 and then convert those vectors into a corresponding matrix and row reduce and your pivotal columns will tell you what vectors are linearly independent.