Second Semester Linear Algebra

[LOannie234]

The vectors u1=(1,-1,2), u2=(-1,2,-1), and u3=(0,2,1) form a basis for C^3.

a) Find the matrix that changes the coordinates in the basis u1,u2,u3 into coordinates in the usual basis.

b) Find the matrix that changes the coordinates in the usual basis into coordinates in the basis u1,u2,u3.

I am completely stuck with this problem so far. All I have done is put the vectors into a matrix and found the inverse of it:
[-4,-1,2;-5,-1,2;3,1,-1]

Any help would be very much appreciated!!

 

[SlipEternal]

When posting here, you can write matrices by using LaTeX. For instance, to write that matrix, you can write:

[tex}\begin{pmatrix}-4 & -5 & 3 \\ -1 & -1 & 1 \\ 2 & 2 & -1 \end{pmatrix}[/tex]

(But I wrote [tex}, you should change the } to a ])

It will appear like this:
\(\displaystyle \begin{pmatrix}-4 & -5 & 3 \\ -1 & -1 & 1 \\ 2 & 2 & -1 \end{pmatrix}\)

Anyway, to answer your question, it is easier to figure out the answer if you have a better idea of what the change of coordinate matrix is actually supposed to do. Essentially, if you are given a vector \(\displaystyle \begin{pmatrix}a\\b\\c\end{pmatrix}\) with coordinates in \(\displaystyle u_1,u_2,u_3\), it means that you have \(\displaystyle au_1+bu_2+cu_3\). This is a linear combination of vectors from your basis. Every linear combination of those three vectors is distinct. Let us rewrite the linear combination:

\(\displaystyle au_1+bu_2+cu_3 = a\begin{pmatrix}1\\-1\\2\end{pmatrix} + b\begin{pmatrix}-1\\2\\-1\end{pmatrix} + c\begin{pmatrix}0\\2\\1\end{pmatrix}\) \(\displaystyle = \begin{pmatrix}a-b\\-a+2b+2c\\2a-b+c\end{pmatrix}\). Written as a linear combination of the standard basis vectors, this would be: \(\displaystyle (a-b)\begin{pmatrix}1\\0\\0\end{pmatrix} + (-a+2b+2c)\begin{pmatrix}0\\1\\0\end{pmatrix} + (2a-b+c)\begin{pmatrix}0\\0\\1\end{pmatrix}\). Thus, this is the matrix you want. \(\displaystyle \begin{pmatrix}1 & -1 & 0 \\ -1 & 2 & 2 \\ 2 & -1 & 1\end{pmatrix}\). So, you rightly put the vectors into a matrix. That is the change of basis matrix from coordinates in \(\displaystyle u_1, u_2, u_3\) to the standard basis.

Then, to change from the standard basis back to \(\displaystyle u_1, u_2, u_3\), you rightly found the inverse of the matrix. So, you already have the answer to both parts.