Write down orthonormal bases for the four fundamental subspaces [ ]

[s3a]

Write down orthonormal bases for the four fundamental subspaces [...]"

Homework Statement

Problem:
Write down orthonormal bases for the four fundamental subspaces of A = matrix([1,2],[3,6]]). (1 and 2 are on the first row whereas 3 and 6 are on the second row.)

Solution:
A = matrix([1,2],[3,6]]) is a 2 by 2 matrix of rank 1. Its row space has basis $v_1$, its nullspace has basis $v_2$, its column space has basis $u_1$, its left null space has basis $u_2$:

Row space: 1/sqrt(5) matrix([1],[2])
Nullspace: 1/sqrt(5) matrix([2],[-1])
Column space: 1/sqrt(10) matrix([1],[3])
Left nullspace: 1/sqrt(10) matrix([3],[-1])

Homework Equations

Gram-Schmidt process (I think)

The Attempt at a Solution

I watched videos on the Gram-Schmidt process but, they involve vectors whereas this involves a matrix, plus the concept with the vectors is still new to me so could someone help me with this super basic problem so that I can get started with the more complex ones please?

Any input would be greatly appreciated!

 

[s3a]

Actually, I figured out the row and column vector parts of this question but, I have no idea how to do the nullspace and left nullspace parts. It's probably a simple extension of the concept but, I don't know what to do. I have a feeling that this problem is really simple so, it should take minimal effort to help so, please do.

 

[Mark44]

Homework Statement

Problem:
Write down orthonormal bases for the four fundamental subspaces of A = matrix([1,2],[3,6]]). (1 and 2 are on the first row whereas 3 and 6 are on the second row.)

Solution:
A = matrix([1,2],[3,6]]) is a 2 by 2 matrix of rank 1. Its row space has basis $v_1$, its nullspace has basis $v_2$, its column space has basis $u_1$, its left null space has basis $u_2$:

Row space: 1/sqrt(5) matrix([1],[2])
Nullspace: 1/sqrt(5) matrix([2],[-1])
Column space: 1/sqrt(10) matrix([1],[3])
Left nullspace: 1/sqrt(10) matrix([3],[-1])

Homework Equations

Gram-Schmidt process (I think)

The Attempt at a Solution

I watched videos on the Gram-Schmidt process but, they involve vectors whereas this involves a matrix, plus the concept with the vectors is still new to me so could someone help me with this super basic problem so that I can get started with the more complex ones please?

Any input would be greatly appreciated!

Actually, I figured out the row and column vector parts of this question but, I have no idea how to do the nullspace and left nullspace parts. It's probably a simple extension of the concept but, I don't know what to do. I have a feeling that this problem is really simple so, it should take minimal effort to help so, please do.

For the nullspace, you're looking at the equation Ax = 0, and finding a basis for this set. For the left nullspace, you're looking at the equation x^TA = 0, and finding a basis for this set. You show answers above. Are these from the back of the book, and you're uncertain how they were found?

BTW, LaTeX provides the nicest presentation of matrices, and it's not that hard. Here's your matrix:
$$ A = \begin{bmatrix} 1 & 2 \\ 3 & 6\end{bmatrix}$$
Right click the matrix above to see my LaTeX script.

For the left nullspace, this is the equation you need to work with:
$$ \begin{bmatrix}x_1 & x_2 \end{bmatrix} \begin{bmatrix}1 & 2 \\ 3 & 6\end{bmatrix} = 0$$