Finding the Kernel and Range of Linear Operators on R3

[Mdhiggenz]

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

 

[jbunniii]

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.

 

[Mdhiggenz]

Perfect explanation. Thank you!