What information can be found in the columns of the transition matrix?

[Lord Anoobis]

Homework Statement

Let $B_1 = {\begin{bmatrix} 1 \\ 1 \\ 1\\ 0 \end{bmatrix}}, {\begin{bmatrix} 1 \\ 1 \\ 0\\ 0 \end{bmatrix}}, {\begin{bmatrix} 0 \\ 0 \\ 1\\ 1 \end{bmatrix}}$ and $B_2 = {\begin{bmatrix} 1 \\ 1 \\ 1\\ 1 \end{bmatrix}}, {\begin{bmatrix} 1 \\ 1 \\ 1\\ -1 \end{bmatrix}}, {\begin{bmatrix} 1 \\ 1 \\ -1\\ 1 \end{bmatrix}}$ be two bases for $span(B_1)$, where the usual left to right ordering is assumed. Find the transition matrix $P$_B1$\to$B2

Homework Equations

The Attempt at a Solution

I'm a bit flummoxed here. All the problems I've dealt with so far have had $n$ $n \times 1$ vectors and were solved by finding inverses. That cannot work here. What would be the first step in solving this?

 

[PeroK]

Homework Statement

Let $B_1 = {\begin{bmatrix} 1 \\ 1 \\ 1\\ 0 \end{bmatrix}}, {\begin{bmatrix} 1 \\ 1 \\ 0\\ 0 \end{bmatrix}}, {\begin{bmatrix} 0 \\ 0 \\ 1\\ 1 \end{bmatrix}}$ and $B_2 = {\begin{bmatrix} 1 \\ 1 \\ 1\\ 1 \end{bmatrix}}, {\begin{bmatrix} 1 \\ 1 \\ 1\\ -1 \end{bmatrix}}, {\begin{bmatrix} 1 \\ 1 \\ -1\\ 1 \end{bmatrix}}$ be two bases for $span(B_1)$, where the usual left to right ordering is assumed. Find the transition matrix $P$_B1$\to$B2

Homework Equations

The Attempt at a Solution

I'm a bit flummoxed here. All the problems I've dealt with so far have had $n$ $n \times 1$ vectors and were solved by finding inverses. That cannot work here. What would be the first step in solving this?

You could express one set of basis vectors in terms of the other basis.

 

[Lord Anoobis]

Would you happen to know how to get formulas to display correctly on an android phone? I'm away from my pc for a spell and I can't see much this way.

 

[member 587159]

The first step: know the relevant definitions and theorems

If you want to find the transition matrix, you have to know what information can be found within it. In general, a transition matrix gives you all the information you need to know to convert coordinates of a certain basis to coordinates relative to another basis. Denote the transition matrix from $B_1$ to $B_2$ with $M$. Do you know what information you can find in the columns of $M$?