Linear algebra question: Orthogonal subspaces

[Mdhiggenz]

Homework Statement

For each of the following matrices, determine a basis for each of the subspaces N(A)

A=[3 4]
[ 6 8]

Homework Equations

The Attempt at a Solution

So reducing it I got [1 4/3]
[0 0]

I know x₂ is a free variable

I set x₂ = to β

and found my N(A)=(-4/3β,β)^T

However the book has simply (-4,3)^T

Am I incorrect?

 

[LCKurtz]

What would have happened if you had set $x_2=3\beta$?

 

[Mdhiggenz]

We get -4B

 

[Infrared]

To find a basis for the nullspace is to find a minimal set of vector(s) that span your nullspace. If the nullspace is $\binom{-4}{3}$ as the book says, then any member of the nullspace can be written as a multiple of $\binom{-4}{3}$. This means that the general form of a vector in N(A) is $\binom{-4β}{3β}$, which is equivalent to what you have. So you have written out the general form of a vector in the nullspace (assuming that your β is a free variable), whereas your book just gave the basis vector.

 

[Mdhiggenz]

Awesome thanks!