# Finding range and kernel of linear transformation

• Ch1ronTL34
In summary, the range for a) is a line, for b) it is a plane, for c) it is a point, and for d) it is a line. The kernel for all of these is the zero subspace.
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!

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.

Ch1ronTL34 said:
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?

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.

## 1. What is a linear transformation?

A linear transformation is a mathematical operation that maps one vector space to another, while preserving the basic algebraic structure of the original space. It is represented by a matrix, and can be used to describe a variety of physical phenomena in fields such as physics, engineering, and economics.

## 2. How do you find the range of a linear transformation?

To find the range of a linear transformation, you need to determine all the possible outputs of the transformation. This can be done by applying the transformation to all possible input vectors and observing the resulting output vectors. The range of the linear transformation is then the set of all output vectors.

## 3. What is the significance of the range of a linear transformation?

The range of a linear transformation represents the span of all possible outputs of the transformation. It is important because it gives insight into the behavior and limitations of the transformation. For example, if the range is limited, it may indicate that the transformation is not able to fully describe the system it is modeling.

## 4. How do you find the kernel of a linear transformation?

To find the kernel of a linear transformation, you need to determine all the input vectors that result in an output of zero. This can be done by solving the system of equations represented by the transformation matrix, where the right-hand side is set to zero. The kernel of the linear transformation is then the set of all input vectors that satisfy this system of equations.

## 5. What is the relationship between the range and the kernel of a linear transformation?

The range and kernel of a linear transformation are related by the fundamental theorem of linear algebra. This theorem states that the dimension of the range plus the dimension of the kernel equals the dimension of the input space. In other words, the range and kernel are complementary and together they span the entire input space.

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