Understanding Basis for Solving Linear Algebra Problems

[ElliottG]

Hey guys

There are so many of these damn "Find a basis" questions and I can't get any of them because we never directly learned how...or she never showed us in class...my final exam is tomorrow.

Here are some examples of questions:

http://184.154.165.18/~devilthe/uploads/1323453294.png

http://184.154.165.18/~devilthe/uploads/1323430492.png

Part D

I have zero idea how they solve these...

I know that a basis is a linearly independent spanning set...but how to solve these questions? No idea.

Can there be more than one basis for a question?

Thanks,
Elliott

 

[Mark44]

A basis is a set of vectors (or matrices in this problem) that is linearly independent and spans the (sub)space. A vector (sub)space can have many bases.

For d, start by playing with the equation that defines the subspace V, using an arbitrary matrix X, where
$$X = \begin{bmatrix}a & b \\ c & d \end{bmatrix}$$

 

[HallsofIvy]

Any polynomial, of degree 2, is of the form $p(x)= ax^2+ bx+ c$. $p'(x)= 2ax+ b$ Requiring that p'(1)= 0 means that 2a+ b= 0 so b= -2a. That is, for any p in this set, $p(x)= ax^2- 2ax+ c= a(x^2- 2)+ c(1)$. Now, what is a basis for that set?

Any 2 by 2 matrix is of the form
$$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$

We require that
$$\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}\begin{bmatrix}a & b \\ c & d\end{bmatrix}= \begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}$$

Multiply those, set corresponding term equal, and see what you get.

 

[ElliottG]

Thanks guys.

Anyway I just did my linear exam today so hopefully I never have to see linear ever again!