Matrix Multiplication and Linear Transformations

[madcap_]

Homework Statement

Homework Equations

The Attempt at a Solution

I'm completely lost on this one.

I think the question is saying matrix A is a representation of a linear transformation, with the a₁₁ and a₂₁ transforming to a₃₁, and so on for the other two columns. But I don't see how you can get that result with a₁₂ being negative. This is all I could come up with after days of looking at this problem, and I'm just going around in circles

As for the product C=BA well that's pretty straightforward.

If someone can point me in the right direction it will be very much appreciated!

 

[lanedance]

so consider A as something like
$$A = \begin{pmatrix} M & \textbf{0} \\ \textbf{0} & 1 \end{pmatrix}$$

with
$$M = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$

and consider the product
$$M = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$

now relate that back to the original matrix A and its operation on and arbitrary x1,x2 and x3

 

[madcap_]

So when you say the product..

$$M = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$

Is x₁ and x₂ equal to m₁₁ and m₁₂?

Which multiplied by the matrix M = $\left[\stackrel{1}{0}\right]$
Edit:- I'm still not getting it. Matrices aren't making a lot of sense to me right now :(

 

[HallsofIvy]

I think lanedance spoke wrong in the last part of his response. I am sure he meant to say
$$A\begin{pmatrix}x_1 \\ x_2\\ x_3\end{pmatrix}= \begin{pmatrix}M & 0 \\ 0 & 1\end{pmatrix}\begin{pmatrix}y_1\\ y_2\end{pmatrix}$$

where
$$y_1= \begin{pmatrix}x_1 \\ x_2\end{pmatrix}$$
and $y_2= x_3$.

Of course, M is now a rotation matrix.

 

[lanedance]

yeah thanks Halls, i didn't mean to have that last equals sign

 

[madcap_]

Ok.. So for A * (x1,x1,x3), x1,x2 are mapped to new points, and x3 remains at the same point. Or with A partitioned to the 2x2 matrix M, and M * (x1,x2) + (0,0,x3) * A, you get exactly the same result right..

Is the partitioning completely arbitrary? I think that's what has been confusing me this whole time

 

[madcap_]

T(x₁,x₂,x₃)=A(x₁,x₂),x₃

$$\begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$

 

[lanedance]

The only reason to look at the partitioning is that x3 is unchanged by the operation in this problem.

So to understand the action of the operator we only need to consider its action on x1 and x2, which is a rotation in that plane, hence why we consider the partitioned matrix.

 

[madcap_]

Ah.. I think it finally makes sense to me now. Thank you!

 

[lanedance]

no worries ;)