Linear algebra: Finding a basis for a space of polynomials

1. Homework Statement
Let
and
are two basis of subspaces
and [PLAIN]http://www.sosmath.com/CBB/latexrender/pictures/69691c7bdcc3ce6d5d8a1361f22d04ac.png. [Broken] Find one basis of http://www.sosmath.com/CBB/latexrender/pictures/38d4e8e4669e784ae19bf38762e06045.png and [PLAIN]http://www.sosmath.com/CBB/latexrender/pictures/fd36d76c568c236aaaad68e084eef495.png. [Broken]

2. Homework Equations
-Vector space
-Basis
-Polynomials

3. The Attempt at a Solution

Could someone explain the method for finding a basis for a space of polynomials.
I know that with
we need to find RREF of an augmented matrix,
and read a basis from matrix, but how to do it with polynomials?

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PeroK
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Do you understand what L and M are? Can you describe these as vector spaces?

HallsofIvy
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The first post says exactly what L and M are- it gives their bases. L is the space of polynomials spanned by $\{1+ t- t^3, 1+ t+ t^2, 1- t\}$ so any vector in L is of the form $a(1+ t- t^3)+ b(1+ t+ t^2)+ c(1- t)= -at^3+ bt^2+ (a+ b- c)t+ (a+ b+ c)$ for any number a, b, and c. M is the space of polynomials spanned by $\{t^3+ t, 2- t^3, 2+ t^3\}$ so any vector is M is of the form $p(t^3+ t)+ q(2- t^3)+ r(2+ t^3)= (p- q+ r)t^3+ pt+ (2q+ 2r)$ for any numbers p, q, and r. Any vector in L+ M is of the form $(-a+ p- q+ r)t^3+ bt^2+ (a+ b- c+ p)t+ (a+ b+ c+ 2q+ 2r)$. Any vector in $L\cap M$ can be written as either of those 2 first forms with -a= p+ q+ r, b= 0, a+ b- c= p, and a+ b+ c= 2q+ 2r. Simplify those.

blue_leaf77
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PeroK
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Looks like this has turned into Halls of Ivy's homework!

HallsofIvy