Let T be projection onto the line y 2x Find the most app

Let   T :² ² be projection onto the line y = - 2x

Find the most appropriate answer from the given choices to each of the following questions.

The domain of T is:

The codomain of T is

The range of T is

Is T onto?

Is T one-to-one?

Locate all points X in ² for which   T(X)= X ?

For any point X in ² T(T(X)) is equal to

Locate all points X in ² for which   T(X) = O (zero vector / origin) ?

The kernel of T is

Solution

Hi, I am Waqar, I have answered your most of the parts and sol will help you to find the rest.

You need to find a matrix A such that Ax=y where x is in R² and y is on the line. One way to do this is to actually calculate the projection of two points onto the line. I would pick (1,0) and (0,1) since they are easy. To find where on the line they are, you just take the scalar projection of each vector onto y=2x. The vector for y= -2x is just (1,-2) and the scalar projection is the dot product normalized for the length, ie

(1,0) . (1,-2) / sqrt(5)

and

(0,1) . (1,-2) / sqrt(5)

Now that you know where you are on the line, you can get the coordinates from Pythagoras.

For (1,0) you have a hypotenuse of 1/sqrt(5). On the line, this means

x² + (-2x)² = 1/5

5x² = 1/5

x=1/5, y=2/5

For (0,1) you have a hypotenuse of -2/sqrt(5).

x² + (-2x)² = 4/5

5x² =4/5

x=2/5, y=4/5

=> For T, y = -2x is Injective (one to One ) on its domain

=> y = -2x is Surjective (onto) R

Also the matrix represtation will be A = | 1/5 2/5 2/5 4/5 |

Regards

Waqar

Happy Chegging