Let T :² ² be projection onto the line y = - 2x Find the most appropriate answer
ID: 3035876 • Letter: L
Question
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
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
Explanation / Answer
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
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