Finding the kernel and range of a tranformation

Noone1982

If L(x) = (x1, x2, 0)^t and L(x) = (x1, x1, x1)^t

What is the kernel and range?

HallsofIvy

You have defined two different functions. Which one are you referring to?

Do you know the DEFINITION of "kernel"

Noone1982

I need to do both. I know the kernel is basically setting to zero, but my book is awful on explaining stuff. I also know the range is basically solving for y and row reducing, but I am foggy on the presentation.

JasonRox

I'll explain it to you, and I want to see you attempt to answer it.

The range is simple. It's just like Calculus, sort of.

For example, the function f(x)=x^2 has a domain of R(-infinite,infinite), and range (0,infinite).

The range is simply all the possible vectors you can obtain from the transformation. If T(x) = (0,0), then all the possible vectors that come "out" of the transformation are in the set {(0,0)}, which is the range.

What's the range of T(x)=(x,y)?

The answer is R^2, which is any vector in the Cartesian Plane.

The kernel of a transformation is the set of vectors that transform into the zero vector. So, for the first one T(x)=(0,0) is all the vectors, since all the them become a zero vector, so the answer is R^2.

What's the kernel of T(x)=(x,y)?

Well, the only possible vector that can transform into a zero vector is the zero vector itself.

Note: I have no idea what you mean by exponent t. Maybe that signifies that it is a transformation... I have no idea.

HallsofIvy

Your book is awful on explaining "setting to zero"?? Not a whole lot to explain is there?

For your first function L(x₁,x₂,x₃))= (x₁, x₂,0).
Set that equal to 0: (x₁, x₂,0)= (0, 0, 0).

What does that tell you about x₁ and x₂? What does it tell you about x₃?

Now the range: what do all possible values of L, that is all vectors of the form (x₁, x₂,0), have in common?

George Jones

It means transpose. For example (5,2,3) is a row matrix (i.e., a 1x3 matrix) and (5,2,3)^t is a column matrix (i.e., a 3x1 matrix).

Regards,
George

JasonRox

I know what transpose means.

HallsofIvy

Then I'm surprised that you don't know that A^T is a standard notation for transpose.

JasonRox

Exactly... capital T.

matt grime

You can't guess that lower and upper case t in an expression x^t or x^T where x is a vector don't both obviously mean transpose?

JasonRox

I didn't think the transpose was necessary.

Anyways, the question isn't about transposes.

matt grime

It's about vectors, and linear maps. What on Earth was it going to be except transpose?