# Matrix and linear transformation

• Apr 25th 2010, 11:29 PM
ulysses123
Matrix and linear transformation
Identify the following matrices as reflections, shears, scalings,rotation or projections for the unit square.

$\left(\begin{array}{cc}1/5&2/5\\2/5&4/5\end{array}\right)
$

When i graph this i get a stright line y=2x.
In addition solving for the mapped values (x' and y') in terms of the inputed values x and y i again get the equation y'=2x'

which implies that the mapped y values are a function of the mapped x values.But i still can't interpret this. It is not a reflection, and it is not a dilation so what type of matrix is this?
• Apr 26th 2010, 03:59 AM
HallsofIvy
You've essentially got the answer! That matrix has determinant 0 and so is not invertible- specifically, as you say, it maps all point in the plane to a single line. What kind of transformation is that? Not a "reflection" surely, nor a rotation. What kind of tranformation maps everything onto a subspace?
• Apr 26th 2010, 08:27 PM
ulysses123
I am guessing its must be a projection onto the line y=2x, because out of all the transformations we have covered in class, this is the only one i havnt really understood. (ie projections).So if you could explain some key points i would appreciate it.

From my understanding a projection maps all points from a vector space onto the same vector space but the image is different.
• Apr 27th 2010, 03:09 AM
HallsofIvy
Quote:

Originally Posted by ulysses123
I am guessing its must be a projection onto the line y=2x, because out of all the transformations we have covered in class, this is the only one i havnt really understood. (ie projections).So if you could explain some key points i would appreciate it.

From my understanding a projection maps all points from a vector space onto the same vector space but the image is different.

Not "onto", "into". A projection maps every point in a vector space onto a subspace of that vector space.

And, yes, this is a projection onto the subspace {(x, 2x)}.