Basis for Image and Kernel of matrix

[boneill3]

Homework Statement

Find an Basis for Image and Kernel of the matrix.
$\[ \left( \begin{array}{ccc}<br /> 2 & 1 & 3 \\<br /> 0 & 2 & 5 \\<br /> 1 & 1 & 1 \end{array} \right)\]$

Homework Equations

The Attempt at a Solution

To find the kernel I solve the equation Ax = 0

I put the matrix in row reduced echelon form which is the identity matrix.
$\[ \left( \begin{array}{ccc}<br /> 1 & 0 & 0 \\<br /> 0 & 1 & 0 \\<br /> 0 & 0 & 1 \end{array} \right)\]$

Therefore its the equation

x = 0
y = 0
z = 0

The kernel basis is just the unit basis, {(1,0,0),(0,1,0),(0,0,1)}

For the image basis I've seen that you can use the pivots of the rref matrix and use the corresponding column vectors of the original Matrix as the image basis.

So is that just

{(2,0,1),(1,2,1),(3,5,1)} ?

 

[boneill3]

Sorry wrong latex code

 

[Billy Bob]

The kernel basis is just the unit basis, {(1,0,0),(0,1,0),(0,0,1)}

No. You found the kernel to be trivial, i.e. {(0,0,0)}. The basis for the kernel, then, is the empty set.

For the image basis I've seen that you can use the pivots of the rref matrix and use the corresponding column vectors of the original Matrix as the image basis.

So is that just

{(2,0,1),(1,2,1),(3,5,1)} ?

This is correct. You could also use {(1,0,0),(0,1,0),(0,0,1)} since the image is all of R3. You might have to write them as column vectors, depending on the conventions your instructor has adopted.

 

[boneill3]

Thanks