Find a spanning set a minimal spanning set for ##P_4##.

[Terrell]

Homework Statement

Find a spanning set for $P_4$. Find a minimal spanning set. Use Theorem 2.7 to show no other spanning set has fewer elements.
Would simply like someone to check my answers as the book I'm using did not provide a solutions manual. Thank you.

Homework Equations

Theorem 2.7: If a finite set $A$={$\alpha_1,\alpha_2,...,\alpha_n$} spans $V$, then every linearly independent set contains at most $n$ elements.

The Attempt at a Solution

spanning set: {$1,x,x^2,x^3,5$}
minimal spanning set: {$1,x,x^2,x^3$}
By theorem 2.7., since {$1,x,x^2,x^3$} is linearly indepdent and spans $P_4$, then any linearly independent set has at most $m$ elements such that $m\leq 4$. However, {$1,x,x^2,x^3$} has 4 elements and is a minimal spanning set $\Rightarrow$ $4\leq m$. Therefore, $m=4$.

 

[mfb]

Shouldn't P₄ have 4-th degree polynomials in it?

To use theorem 2.7 you have to show (or at least say) that your spanning set is linear independent.

 

[Terrell]

If I wanted to show that the set is linearly independent, do i need to show explicitly that each element can't be a linear combination of the other elements in the set or is there a shorter method? My book defined $P_n$ to be of degree $\leq n-1$.

 

[mfb]

If I wanted to show that the set is linearly independent, do i need to show explicitly that each element can't be a linear combination of the other elements in the set or is there a shorter method?

You can probably get away with just saying it is linear independent. To prove it formally, you can show that the only way to get the null vector is zero for all coefficients.

My book defined $P_n$ to be of degree $\leq n-1$.

Okay, fine.

 

[Terrell]

o prove it formally, you can show that the only way to get the null vector is zero for all coefficients.

can solving it using gauss-jordan elimination in matrix form be considered formal? thanks!

 

[mfb]

That will work. It is trivial for the spanning set you chose, of course.

 

[Mark44]

Shouldn't P₄ have 4-th degree polynomials in it?

Authors of linear algebra textbooks aren't consistent in this notation. Two of my books define $P_n$ as the space of polynomials of degree less than n. Another defines this as polynomials of degree less than or equal to n.