Finding the Kernel and Range of Linear Operators on R3

In summary, the kernel of the linear operator L(X)=(x1,x1,x1)T on R3 is the set of vectors of the form (0,x2,x3), and it is a 2-dimensional subspace because the vectors (0,1,0) and (0,0,1) form a basis for this subspace.
  • #1
Mdhiggenz
327
1

Homework Statement



Determine the kernel/range of each of the following linear operators on R3

L(X)=(x1,x1,x1)T

Homework Equations


The Attempt at a Solution



So first thing I did was create a 3x1 matrix filled with ones.

I equaled it to zero and found x1=0 to be a solution. However I'm not quite sure how they come up with the following answer.

(0,x2,x3). Also why would it be a 2 dimensional subspace? Would it be due to x1 being zero?

Thanks
 
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  • #2
Mdhiggenz said:

Homework Statement



Determine the kernel/range of each of the following linear operators on R3

L(X)=(x1,x1,x1)T

So first thing I did was create a 3x1 matrix filled with ones.

I equaled it to zero and found x1=0 to be a solution. However I'm not quite sure how they come up with the following answer.

(0,x2,x3). Also why would it be a 2 dimensional subspace? Would it be due to x1 being zero?
x1 = 0 if and only if the vector is of the form (0,x2,x3). The subspace spanned by vectors of this form has dimension 2 because for example {(0,1,0), (0,0, 1)} is a basis.
 
  • #3
Perfect explanation. Thank you!
 

FAQ: Finding the Kernel and Range of Linear Operators on R3

1. What is the kernel of a linear transformation?

The kernel of a linear transformation is the set of all vectors in the domain that are mapped to the zero vector in the codomain. In other words, it is the set of all inputs that result in an output of zero.

2. How is the kernel related to the range of a linear transformation?

The kernel and range of a linear transformation are complementary subspaces. This means that any vector in the domain can be decomposed into a sum of two vectors, one in the kernel and one in the range.

3. Can the kernel and range of a linear transformation be empty?

Yes, it is possible for the kernel and range of a linear transformation to be empty. This occurs when the linear transformation is injective, meaning that each input in the domain corresponds to a unique output in the codomain. In this case, the only vector in the kernel is the zero vector, and the range is equal to the entire codomain.

4. How can the kernel and range of a linear transformation be used in applications?

The kernel and range of a linear transformation can be used to solve systems of linear equations, find eigenvalues and eigenvectors, and analyze the behavior of a system in mathematical models. They are also important in fields such as computer graphics, machine learning, and signal processing.

5. Is the kernel always a subspace of the domain?

Yes, the kernel is always a subspace of the domain. This is because the zero vector in the domain is always mapped to the zero vector in the codomain, and the kernel must contain the zero vector by definition. Additionally, the kernel must be closed under addition and scalar multiplication, which are properties of subspaces.

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