# Linear Algebra - Change of basis matrices and RREF question what in the world?

1. Feb 6, 2010

### zeion

Linear Algebra - Change of basis matrices and RREF question what in the world??

1. The problem statement, all variables and given/known data

Suppose the linear transformation T: P3 -> P2, over R has the matrix

$$A = \begin{bmatrix}1&2&0&0\\0&1&2&1\\1&1&1&1 \end{bmatrix}$$
relative to the standard bases of P3 and P2.

Find bases A' pf P3 and B' of P2 such that the matrix A' of T relative to A' and B' is the reduced row-echelon form for A.

2. Relevant equations

3. The attempt at a solution

Okay, so the relation is:
$$T_{B'A'} = I_{B'B}T_{BA}I_{AA'}$$

Where A is the standard basis for for P3 and B is the standard basis for P2.

I also have the relation:

R = VA, where R is the RREF of A and V is an invertible matrix that maps A to R.

If I take the basis A' of A to be equal to A, then
$$I_{AA'} = _{4x4}$$

and $$I_{B'B} = V$$

Then I form an augmented matrix to find V:

$$\begin{bmatrix}1&2&0&0&|&1&0&0\\0&1&2&1&|&0&1&0\\1&1&1&1&|&0&0&1\end{bmatrix}$$

which row reduces to

$$\begin{bmatrix}1&0&0&2/3&|&-1/3&2/3&4/3\\0&1&0&-1/3&|&2/3&-1/3&-2/3\\0&0&1&2/3&|&-1/3&1/3&1/3\end{bmatrix}$$

Then V is the augmented part..
After this I will find the inverse of V, then find B'.. but it doesn't look right, and the final answer I got was wrong ~_~

Can someone confirm if this is correct track or if I made a mistake somewhere? Thanks.

2. Feb 7, 2010

### vela

Staff Emeritus
Re: Linear Algebra - Change of basis matrices and RREF question what in the world??

It looks like you just made an arithmetic error. V12 and V22 have the wrong sign.

3. Feb 7, 2010

### zeion

Re: Linear Algebra - Change of basis matrices and RREF question what in the world??

Okay this really angers me I tried redoing it so many times and still can't get the right answer. Matrix operations really makes me angry.

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