# Thread: Two Linear Transformation Problems

1. ## Two Linear Transformation Problems

I think I've got a rather glaring hole in my understanding of this material. I've read and reread the text, watched the MIT Open Courseware lectures, and beat my head against this material for quite a while now. Any help would be much appreciated. Without further ado...

1) Define a linear transformation $\displaystyle L: P_{3} \rightarrow P_{2}$ by $\displaystyle L(f(x) = f'(x) + \int_0^6 \! f(x) \, \dif x$. Consider ordered bases $\displaystyle B = {1, x, x^2}$ and $\displaystyle C = {1, x}$ for $\displaystyle P_{3}$ and $\displaystyle P_{2}$ respectively.

a) Find the representation matrix $\displaystyle A$ of $\displaystyle L$ with respect to the given bases $\displaystyle B$ and $\displaystyle C$.
b) Express $\displaystyle [L(a_{0} + a_{1}x + a_{2}x^2)]_C$ in terms of $\displaystyle a_{0}, a_{1}, a_{2}$

2) Let $\displaystyle \mathbb{R}^{2 x 2}$ be the vector space of all 2 x 2 matrices (over $\displaystyle \mathbb{R}$. Consider
$\displaystyle E_{11} = \begin{array}{|cc|}1&0\\0&0\end{array}$, $\displaystyle E_{12} = \begin{array}{|cc|}0&1\\0&0\end{array}$, $\displaystyle E_{21} = \begin{array}{|cc|}0&0\\1&0\end{array}$, $\displaystyle E_{22} = \begin{array}{|cc|}0&0\\0&1\end{array}$; and $\displaystyle M = \begin{array}{|cc|}3&4\\5&6\end{array}$.

It is known that $\displaystyle E = E_{11}, E_{12}, E_{21}, E_{22}$ is an ordered basis for $\displaystyle \mathbb{R}^{2 x 2}$. Define $\displaystyle L: \mathbb{R}^{2 x 2} \rightarrow \mathbb{R}^{2 x 2}$ by $\displaystyle L(x) = Mx$ for all $\displaystyle x$ = $\displaystyle \begin{array}{|cc|}x_11&x_12\\x_21&x_22\end{array} \in \mathbb{R}^{2 x 2}$.

Edit: Couldn't get the format right for that x matrix. Is should be x_{11}, x_{12}, x_{21}, x_{22}.

a) Find $\displaystyle [M]_E$, the coordinate vector of $\displaystyle M$ with respect to the ordered basis $\displaystyle E$.
b) Is L a linear transformation?
c) Find the representation matrix of L with respect to $\displaystyle E$, if L is a linear transformation.

For 1), I found $\displaystyle L(1) = 6, L(x) = 19, and L(x^2)= 2x+72$. Then I tried to set up a matrix with the bases of C and those values. I interpreted the bases of C as the standard bases (1, 0), (0, 1). I really don't think that's right, so I'm kind of stuck there.

For 2), I can't even make headway on the first part. I set up the problem like I would to find a coordinate vector, but it comes out as a 2 x 10 matrix. Trying to account for the zero columns still leaves me with a 2 x 6. I believe that this is a linear transformation, but as I said, I'm quite stumped.

Any help would be very much appreciated.

2. ## Re: Two Linear Transformation Problems

for 2a):

Since $\displaystyle M = 3E_{11}+4E_{12}+5E_{21}+6E_{22}$,

the coordinate vector of $\displaystyle M$ with respect to the ordered basis $\displaystyle E$ is $\displaystyle [M]_E = (3,4,5,6).$

for 2b):

$\displaystyle L(x+y) = M(x+y) = Mx+My = L(x)+L(y)$,
$\displaystyle L(\lambda x) = M(\lambda x) = \lambda Mx = \lambda L(x)$,

$\displaystyle \forall \lambda \in \mathbb{R}$ and $\displaystyle \forall x,y \in \mathbb R^{2x2}$ so $\displaystyle L$ is a linear transformation.

for 2c):

$\displaystyle L(E_{11}) = ME_{11}=$$\displaystyle \left(\begin{array}{cc}3&4\\5&6\end{array}\right) \left(\begin{array}{cc}1&0\\0&0\end{array}\right) = \left(\begin{array}{cc}3&0\\5&0\end{array}\right)= 3E_{11}+0E_{12}+5E_{21}+0E_{22}$

so the first column of representation matrix of $\displaystyle L$ with respect to $\displaystyle E$ is $\displaystyle \left(\begin{array}{cc}3\\ 0\\ 5\\ 0\end{array}\right)$.

In a similar way you can find the rest three columns.

3. ## Re: Two Linear Transformation Problems

Thank you very much. I feel incredibly dense for not having seen$M = 3E_{11}+4E_{12}+5E_{21}+6E_{22}$, especially since I had been staring at $\displaystyle M = c_{1}E_{11} + c_{2}E_{12} + c_{3}E_{21} + c_{4}E_{22}$ for so long trying to make sense of a coordinate vector in terms of a 2 x 2 matrix.

Once I read that comment the third part fell into place.

As for problem 1, it turns out I was correct. Evaluating the linear transformation with the given ordered bases of $\displaystyle P_{3}$, and then putting those in an augmented matrix with the ordered bases of $\displaystyle P_{2}$ (interpreted as (1, 0) and (0,1)) gives the matrix representation.

Thanks very much for the help.