Finding range and kernel of linear transformation

[Ch1ronTL34]

Find the range and kernel of:
a) T(v1,v2) = (v2, v1)
b) T(v1,v2,v3) = (v1,v2)
c) T(v1,v2) = (0,0)
d) T(v1,v2) = (v1, v1)

Unfortunately the book I'm using (Strang, 4th edition) doesn't even mention these terms and my professor isn't helpful. My professor said:

"Since range and kernel are subspaces of R^2 (in this problem) you need not give a basis but, rather, simply describe the subspace (i.e. plane, line, zero subspace)"

I don't really know what I'm supposed to find. Let's look at d)

I constructed a 2 x 2 matrix:

[1 0][v1] [v1]
[1 0][v2] =[v1]

Up to this point, I'm not sure what to do or if I'm doing this problem right...I'm stuck, please help!

 

[StatusX]

So you don't know what "range" and "kernel" mean? Then why were you given this assignment? Start by looking up what those terms mean. Mathworld or wikipedia are good for this.

 

[HallsofIvy]

Find the range and kernel of:
a) T(v1,v2) = (v2, v1)
b) T(v1,v2,v3) = (v1,v2)
c) T(v1,v2) = (0,0)
d) T(v1,v2) = (v1, v1)

Unfortunately the book I'm using (Strang, 4th edition) doesn't even mention these terms and my professor isn't helpful.

Strang certainly does define both range and kernel! Have you tried looking in the index?

My professor said:

"Since range and kernel are subspaces of R^2 (in this problem) you need not give a basis but, rather, simply describe the subspace (i.e. plane, line, zero subspace)"

I don't really know what I'm supposed to find. Let's look at d)

I constructed a 2 x 2 matrix:

[1 0][v1] [v1]
[1 0][v2] =[v1]

Up to this point, I'm not sure what to do or if I'm doing this problem right...I'm stuck, please help!

Your professor told you exactly what to do but since you have already decided he is "unhelpful", perhaps you didn't listen.
If (x,y) is a point in the plane, then T(x,y)= (x,x). The range is the set of all points (x,y) such that (x,y)= (x,x). That means y= x. Now, describe that: is that a line, a plane, a point? The kernel is the set of all point (x,y) such that T(x,y)= (0,0). Again, T(x,y)= (x,x)= (0,0). That means x= 0. Can you describe that?

 

[thepasswordis]

Quote: Your professor told you exactly what to do but since you have already decided he is "unhelpful", perhaps you didn't listen.

Or perhaps the professor's explanation is not adequate to connect what the student already knows to what is new to learn. Why is it that experts always seem to think that if a student just "reads" what the words mean, then they MUST get how to do the problem? You are assuming that it is that easy just based on the fact that you already understand the material. To someone who doesn't, it is not as easy as just looking up a definition. And why do theoretical math textbooks invariably like solving very easy, theoretical problems or "proving" proofs but often do not show a student how to do a practical problem? Maybe experts and academics in the sciences can get to work solving those problems while we students struggle with teaching ourselves this material.