# Math Help - Linear Algebra Question.

1. ## Linear Algebra Question.

Let A and B be vector spaces, (which turn out to be real btw) that have ordered bases alpha=(a1,a2,a3) and beta = (b1,b2,b3,b4). The matrix T: A->B is: 2 9 7 4 5 7 9 4 3 2 8 3 work out the alpha coord of a1+a2+a3 and hence or otherwise find T(a1+a2+a3) relative to b1,b2,b3,b4

We have vector spaces $A,B$, with bases $\alpha=(a_{1},a_{2},a_{3})$ and $\beta=(b_{1},b_{2},b_{3},b_{4})$. We then have a matrix $T= \left(\begin{array}
2 & 9 & 7\\
4 & 5 & 7\\
9 & 4 & 3\\
2 & 8 & 3
\end{array}\left)
$

The matrix isn't working in the Latex editor, so let's just go on.

OK.
Tell me everything you know about vector space bases.

3. The alpha coordinates $(\alpha_1,\alpha_2,\alpha_3)$ represent the vector $\alpha_1a_1+\alpha_2a_2+\alpha_3a_3$

Can you do the first part now?

4. Originally Posted by SilenceInShadows
Let A and B be vector spaces, (which turn out to be real btw) that have ordered bases alpha=(a1,a2,a3) and beta = (b1,b2,b3,b4). The matrix T: A->B is: 2 9 7 4 5 7 9 4 3 2 8 3 work out the alpha coord of a1+a2+a3 and hence or otherwise find T(a1+a2+a3) relative to b1,b2,b3,b4
Well since your $T \equiv \begin{pmatrix}
2 & 9 & 7\\
4 & 5 & 7\\
9 & 4 & 3\\
2 & 8 & 3
\end{pmatrix}$
and in terms of co-efficients $a_1 + a_2 +a_3$ translates to $(1,1,1)$, finding $T(a_1+a_2+a_3)$ amounts to computing $\begin{pmatrix}
2 & 9 & 7\\
4 & 5 & 7\\
9 & 4 & 3\\
2 & 8 & 3
\end{pmatrix}\begin{pmatrix}
1\\
1\\
1
\end{pmatrix}$

Finally this will give you a 4 x 1 vector. These are the coefficients of b_1,b_2,b_3,b_4 in the linear combination of the vector in that order.

Originally Posted by bleesdan

We have vector spaces $A,B$, with bases $\alpha=(a_{1},a_{2},a_{3})$ and $\beta=(b_{1},b_{2},b_{3},b_{4})$. We then have a matrix $T= \left(\begin{array}
2 & 9 & 7\\
4 & 5 & 7\\
9 & 4 & 3\\
2 & 8 & 3
\end{array}\left)
$

The matrix isn't working in the Latex editor, so let's just go on.

OK.
Tell me everything you know about vector space bases.
Hey bleesdan,
Use \begin{pmatrix}2 & 9 & 7\\4 & 5 & 7\\9 & 4 & 3\\2 & 8 & 3\end{pmatrix} to get $\begin{pmatrix}
2 & 9 & 7\\
4 & 5 & 7\\
9 & 4 & 3\\
2 & 8 & 3
\end{pmatrix}$

5. Thanks for your help, i have further questions; its most whats being asked of me that I cant figure.

Ive been asked to write down the change matrix been beta to alpha. Where beta is: (4i+6j, 7i+2j, 3i+5j+9k) and alpha has the standard basis. Am I just being stupid and the change matrix is?

4 6 0
7 2 0
3 5 9

or do I need to do something to it? or?

next part is find the matrix which represents
4 5 6
7 3 5
3 5 8
relative to beta

6. Ive been asked to write down the change matrix been beta to alpha. Where beta is: (4i+6j, 7i+2j, 3i+5j+9k) and alpha has the standard basis. Am I just being stupid and the change matrix is?

4 6 0
7 2 0
3 5 9
correct. Well done.

next part is find the matrix which represents
4 5 6
7 3 5
3 5 8
relative to beta
I'm not sure what this part is asking, sorry.