Change of Basis.

**pearlyc** · May 10th 2008, 10:43 PM

Okay, I did most of the parts in this question, just wanted you guys to check if it's the correct answer and teach me on the parts that I don't know how to do! Thanks a lot guys.

Question :
Let S denote the standard basis for $\mathbb{R}^2$ and B = { $[\begin{matrix} 1 \\ 2\end{matrix}]$ , $[\begin{matrix} -1 \\ 0\end{matrix}]$ } be another basis.

(a) write down the change of basis matrix P B->S, from the basis B to the basis S.

P B->S = $[\begin{matrix} 1 & -1 \\ 2 & 0\end{matrix}]$

(b) Hence find the change of basis matrix P S->B, from the basis S to the basis B.

P S-> B = inverse of (P B->S)
= inverse of $[\begin{matrix} 1 & -1 \\ 2 & 0\end{matrix}]$
= 1/2 $[\begin{matrix} 0 & 1 \\ -2 & 1\end{matrix}]$

(c) Find $[x]_B$ if x = (4,-1)

I don't know how to do this one, can you guys help me out?

(d) Show that $T : \mathbb{R}^2 -> \mathbb{R}^2$ defined by :

$T(x,y) = (-x, + 2y, 3y)$

Let x = [ x1, x2], y = y1, y2]

T(x+y)= T(x1 + y1, x2 + y2) = (-x1 + 2y1, 3y1)
= ( -x1, 0.x2) + (2y1, 3y2)
= T(x1,x2) + T(y1,y2)

T(ax) = T(ax1, ax2)
= (-ax1, a.0)
= a (-x1, 0)
= aT(x)

(e) Find the matrix representation of T with respect to the standard basis $[T]_s$

$A_T$ = $[\begin{matrix} -1 & 2 \\ 0 & 3\end{matrix}]$

(f) Use your answers from above to find the matrix representation of T with respect to the basis B, $[T]_B$

$T[b] = P S->B . [T]_S . P B-> S$
= $[\begin{matrix} 3 & 0 \\ 0 & -1\end{matrix}]$

(g) Find $[T(x)]_B$ where $[x]_B$ = $[\begin{matrix} -3 \\ 5\end{matrix}]$

$[Tx]_B$ = [tex][T]_B[x]_B = $[\begin{matrix} -9 \\ 5\end{matrix}]$

**Reckoner** · May 11th 2008, 01:00 AM

Hi, pearlyc!

Originally Posted by pearlyc

Let S denote the standard basis for $\mathbb{R}^2$ and B = { $[\begin{matrix} 1 \\ 2\end{matrix}]$ , $[\begin{matrix} -1 \\ 0\end{matrix}]$ } be another basis.

(a) write down the change of basis matrix P B->S, from the basis B to the basis S.

P B->S = $[\begin{matrix} 1 & -1 \\ 2 & 0\end{matrix}]$

Correct!

Originally Posted by pearlyc

(b) Hence find the change of basis matrix P S->B, from the basis S to the basis B.

P S-> B = inverse of (P B->S)
= inverse of $[\begin{matrix} 1 & -1 \\ 2 & 0\end{matrix}]$
= 1/2 $[\begin{matrix} 0 & 1 \\ -2 & 1\end{matrix}]$

Correct!

Originally Posted by pearlyc

(c) Find $[x]_B$ if x = (4,-1)

I don't know how to do this one, can you guys help me out?

What's the trouble? You have a vector with coordinates relative to the standard basis, and need to find the coordinates relative to the basis $B$ . Use your transition matrix!

Originally Posted by pearlyc

(d) Show that $T : \mathbb{R}^2 -> \mathbb{R}^2$ defined by :

$T(x,y) = (-x, + 2y, 3y)$

Let x = [ x1, x2], y = y1, y2]

T(x+y)= T(x1 + y1, x2 + y2) = (-x1 + 2y1, 3y1)
= ( -x1, 0.x2) + (2y1, 3y2)
= T(x1,x2) + T(y1,y2)

T(ax) = T(ax1, ax2)
= (-ax1, a.0)
= a (-x1, 0)
= aT(x)

Are you being asked to prove that $T$ is a linear transformation?

Your work is completely wrong. I have highlighted your errors in red.

Follow the definition of $T$ , and don't confuse the different sets of coordinates that you have in use. For example, here is how you could do the first part:

$T(x,\ y) = (-x + 2y,\ 3y)$

$\text{Let }\textbf{u} = (u_1,\ u_2)\text{ and }\textbf{v} = (v_1,\ v_2)$

$\text{So, }T(\textbf{u} + \textbf{v})$

$=T(u_1 + v_1,\ u_2 + v_2)$

$=\left( -(u_1 + v_1) + 2(u_2 + v_2),\ 3(u_2 + v_2) \right)$

$=\left( -u_1 - v_1 + 2u_2 + 2v_2,\ 3u_2 + 3v_2 \right)$

$=\left( -u_1 + 2u_2 - v_1 + 2v_2,\ 3u_2 + 3v_2 \right)$

$= (-u_1 + 2u_2,\ 3u_2) + (-v_1 + 2v_2,\ 3v_2) $

$= T(u_1,\ u_2) + T(v_1,\ v_2) $

$= T(\textbf{u}) + T(\textbf{v}) $

See if you can prove the second property now.

Originally Posted by pearlyc

(e) Find the matrix representation of T with respect to the standard basis $[T]_s$

$A_T$ = $[\begin{matrix} -1 & 2 \\ 0 & 3\end{matrix}]$

Correct!

Originally Posted by pearlyc

(f) Use your answers from above to find the matrix representation of T with respect to the basis B, $[T]_B$

$T[b] = P S->B . [T]_S . P B-> S$
= $[\begin{matrix} 3 & 0 \\ 0 & -1\end{matrix}]$

Looks good to me.

Originally Posted by pearlyc

(g) Find $[T(x)]_B$ where $[x]_B$ = $[\begin{matrix} -3 \\ 5\end{matrix}]$

$[Tx]_B$ = [tex][T]_B[x]_B = $[\begin{matrix} -9 \\ 5\end{matrix}]$

Check your multiplication here.

**pearlyc** · May 11th 2008, 04:10 AM

Thanks for your help and effort to check!

I still don't know how to approach part (c) though, haha.

**Moo** · May 11th 2008, 04:14 AM

Hello,

Originally Posted by pearlyc

Thanks for your help and effort to check!

I still don't know how to approach part (c) though, haha.

If P is the transition matrix between basis E and F, then vector X(x1, x2) in E will have the following coordinates in F : PX

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