Subspace of P3, linearly independence?

  • Thread starter csc2iffy
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  • #1
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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[itex]\in[/itex]ℝ

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

Answers and Replies

  • #2
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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[itex]\in[/itex]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.
 

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