Linear Transformation with a Matrix

[Siann122]

Homework Statement

Write down the 2 × 2 matrix that represents the following linear transformation of
the plane. Also draw the image of the (first quadrant) unit square 1.

T(x, y) = (2x +6y, x + 3y).

Homework Equations

T(x, y) = (2x +6y, x + 3y).

The Attempt at a Solution

So the previous four questions were as follows:
(a) A dilation with horizontal dilation factor 2 and vertical dilation factor 1/2.
(b) A vertical shear with factor −1/2.
(c) A rotation of 45◦ anticlockwise.
(d) A reflection across the line at angle −30◦ to the X-axis.

I've done all of these without a problem, but I'm not exactly sure what method you use to actually figure out this one? What exactly is the transformation that's happening? I'm stuck as to what the equation actually means, and I should be okay from there.

 

[dx]

What is the result of applying the matrix

|a b|
|c d|

to the vector (x, y)? What must a, b, c and d be in order to get (2x +6y, x + 3y)?

 

[HallsofIvy]

Another way: note that
$$\begin{bmatrix}a & b \\ c & d \end{bmatrix}\begin{bmatrix}1 \\ 0 \end{bmatrix}= \begin{bmatrix}a \\ c\end{bmatrix}$$
and
$$\begin{bmatrix}a & b \\ c & d \end{bmatrix}\begin{bmatrix}0 \\ 1 \end{bmatrix}= \begin{bmatrix}b \\ d\end{bmatrix}$$

That is, the columns are the results of applying the transformation to the basis vectors.

So, what are T(1, 0) and T(0, 1)?

 

[jambaugh]

There are two ways of looking at this and you should practice "seeing" both ways. The first as mentioned is seeing what a given linear transformation does to the standard basis, understanding the linearity part of linear transformation, and seeing how you expand a general vector (x,y) for example, as a linear combination of basis vectors.

The other way is to understand the decomposition of the linear transformation into certain types.
You can scale a given direction (dilation), you can add a multiple of one direction to another (shear) and you can rotate directions. If you can decompose the linear transformation expressed in component form into a sequence of such actions you can then write down the product of the matrices and multiply it out.

E.g. break up T into a sequence of mappings: (x,y)--> (x,3y) --> (x+3y,x+3y) --> (2x+6y,x+3y).
(note this method requires you be a bit more clever and so is less systematic. There are multiple paths to your goal but it utilizes and thus reinforces a deeper level of understanding.)

You can b.t.w. be systematic with this, look at the resolution of row operations on one matrix as left multiplication by another. [This equivalence is a.) important to understand the validity of row operations in yielding solutions to systems of equations, and b.) integral to the method of finding the inverse of a matrix by performing parallel row ops on both the original matrix and the identity matrix.] Then if you can work out how to perform row ops on a column vector to get the resulting transform you get the sequence of matrices whose product is the transformation matrix.

 

[Siann122]

Alright, so by my logic a unit square is created through the following matrix:

1 0
0 1

And if I turn that above formula into a matrix, I will get the following:

{2 6} {x}
{1 3} {y}

Is my logic sound? Does that matrix represent it or am I not quite getting it still?

 

[dx]

That is the correct matrix that represents the linear transformation.

To find out what happens to the unit square, apply the matrix to the vectors (0, 1) and (1, 0)

 

[Siann122]

Thanks so much for your help guys.