# Kernels and Images

#### yukcream

I just start learning Image and Kernel of Linear Tranfsormation, I find it very confuseing that what is the relationship between span, image and kernel in Transformation T= Ax; x=vector.
Therefore I don't know how to describe the image and kernels of the transformation in :
Q1 : Reflection in the line y= x/3 in R-2
Q2 : Orthogonal projection onto the plane x+ 2y +3z = 0 in R-3

Can anyone help me?

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#### matt grime

Homework Helper
The kernel is the set of vectors mapped to the zero vector. How many vectors are mapped to 0 by any reflection? The image is the set of vectors mapped to. given any vector in R^2 can you see how to find out if it has something mapping to it under the reflection? (it may help to realize that if you do any reflection twice you end up with the identity map).

now try applying the same thinking to the projection. what vectors are mapped to the zero vector? what is the image? obvisouly the image must be a subset of the plane you project into, but which subset?

#### yukcream

matt grime said:
The kernel is the set of vectors mapped to the zero vector. How many vectors are mapped to 0 by any reflection? The image is the set of vectors mapped to. given any vector in R^2 can you see how to find out if it has something mapping to it under the reflection? (it may help to realize that if you do any reflection twice you end up with the identity map).

now try applying the same thinking to the projection. what vectors are mapped to the zero vector? what is the image? obvisouly the image must be a subset of the plane you project into, but which subset?
So So confusing ~~>,< Not quite understand what you mean could you mind elaborate more?

#### matt grime

Homework Helper
no, i won't elaborate more. you should try and understand more (maths is not easy, dont' expect to be spoonfed the answer, it will take you some time to understand the definitions. get used to it). as it is the answer follows easily from the definition of kernel. what is the set of vectors mapped to the zero vector?

this undoubtedly sounds harsh (and i'm sure there are people willing to just tell you the answer but that is wrong).

imagine you are doing french. i tell you the endings all regular verbsd take. then i tell you to work out the french for 'i like' given that aimer is a regular verb. would you expect me to spoon feed the answer as j'aime? no, it is up to you to work it out. maths is no different. you have been told that the kernel is the set of vectors sent to 0 so find out what this is.

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#### Jimmy Snyder

Before you work on Q1, I suggest you work on this related problem.

Q0 : Reflection in the line x = 0 (the y-axis) in R-2.

Reflection in this line means that the point whose coordinates are (x, y) will be sent to the point whose coordinates are (-x, y). For instance: (2, 4) will be sent to (-2, 4).

What will (5, 8) be sent to?
What will (-4, -6) be sent to?
What will be sent to (-9, 17)?
What will be sent to (5, -12)?

And now, the \$64,000 question:
What will be sent to (0, 0)?

I will get back to you once you have answered these questions.

#### yukcream

matt grime said:
no, i won't elaborate more. you should try and understand more (maths is not easy, dont' expect to be spoonfed the answer, it will take you some time to understand the definitions. get used to it). as it is the answer follows easily from the definition of kernel. what is the set of vectors mapped to the zero vector?

this undoubtedly sounds harsh (and i'm sure there are people willing to just tell you the answer but that is wrong).

imagine you are doing french. i tell you the endings all regular verbsd take. then i tell you to work out the french for 'i like' given that aimer is a regular verb. would you expect me to spoon feed the answer as j'aime? no, it is up to you to work it out. maths is no different. you have been told that the kernel is the set of vectors sent to 0 so find out what this is.
Thanks for your remind on how to learning maths.
what you are talking is just theory, and denfinion all of it I have already got from a book no need to copy it once again for me. I wnat to understand the "application" just, can you tell me the answer. of the question:
Q1 : Reflection in the line y= x/3 in R-2
Q2 : Orthogonal projection onto the plane x+ 2y +3z = 0 in R-3

#### neurocomp2003

go step by step and do what grims said...
ther kernal-->solve the reflections that map O

all else www.mathworld.com, your textbook should ahve examples...if not get a new textbook

#### matt grime

Homework Helper
no, i just telling you the answer isn't gonig to help you. draw a picture if it helps you to visualize what is going on in a refelction, any reflection, in a line through the origin. (heck, you can just write out the matrix of the linear map and find its determinant to see that it is an invertible map, whcih incidentally has just told you what the kernel and image are).
learning maths is practical, reading a statement doesn't do anything as you have found out. what have you done in order to try and see what is going on? have you drawn the plane and drawn in the line of reflection? have you thought about how it transforms vectors? What does the map send a point (x,y) or (x,y,z) in the second case to? so when does it send x,y,z to zero? you don't need to work out any equations, you can do it from looking at the picture and thinking geometrically.

or is the problem that you don't know what the maps really do? i project R^3 onto a plane orthogonally, you undertstand that any point in R^3 is of the form w+kn where w is in the plane and n is a unit normal vector orthoginal to the plane, k a scalar. so what does porjection into the plane do to w+kn?

"Kernels and Images"

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