Let U be the subspace of R4 given by:
U = nullspace of the matrix
[0 0 2 3 ]
[0 -3 -2 -2 ]
Find a spanning set for U.
Any and all help will be awesome.
Not sure how to start at all
Cheers
The first thing to do is to find out what the nullspace of your matrix actually is - can you find a general form that these vectors take?
Can you think of a spanning set for this vector space? For instance, if they were of the form $\displaystyle \{(2a,5b, 6c,d): a, b, c, d \in \mathbb{F} \}$ then your space would be spanned by the vectors $\displaystyle (2,0,0,0)$, $\displaystyle (0,5,0,0)$, $\displaystyle (0,0,6,0)$ and $\displaystyle (0,0,0,1)$.
I hope that that helps...
@Swlabr: Few questions please
1. In your example $\displaystyle \{(2a,5b, 6c,d): a, b, c, d \in \mathbb{F} \}$ what is the relevance of 2,5,6? Isn't it just same as $\displaystyle \{a,b,c,d): a, b, c, d \in \mathbb{F} \}$
2. Nevertheless, is there a good/structured approach to find dimension and basis of a space given by something like $\displaystyle \{a,2a,a+b,b): a, b \in \mathbb{F} \}$
Yes. Yes it is. No excuses, other than to point that I was still correct...I mean, 4+2=80712/13452...
Yes. The answer to your problem is either one or two, but is clearly two as the entries in the 1st and 4th positions are independent of one another. It is no more than the number of variables given, but can be less if they cancel out with one another.2. Nevertheless, is there a good/structured approach to find dimension and basis of a space given by something like $\displaystyle \{a,2a,a+b,b): a, b \in \mathbb{F} \}$
$\displaystyle (1, 2, 1, 0)$ and $\displaystyle (0, 0, 1, 1)$ span the space you gave.
In the vectors you gave you had an $\displaystyle a$ and a $\displaystyle b$, so two variables. Say you had a subspace of vectors where you can write them like you did but with $\displaystyle n>0$ variables. Then you can easily split your general form into $\displaystyle n$ vectors each with precisely one variable. In each vector you can take out this variable as a common factor, and the set of all of these vectors (the ones with the variables removed) forms a spanning set. You may be able to reduce this spanning set, if not then it is a basis.
Does that make sense?
Note also that a vector space cannot have a basis of order greater than the length of the vector...
Thanks. I need to think this carefully though. I was also wondering if it has to do something with the fact that each component is a linear combination of the variables. What happens if we relax that - for e.g. {a^2,b^3,a+b^-1,b} (I have not checked if it is a sub-space.)
I presume you mean the matrix
$\displaystyle \begin{bmatrix}0 & 0 & 2 & 3 \\ 0 & -3 & -2 & -2\end{bmatrix}$
The kernel is, by definition, the set of vectors $\displaystyle \begin{bmatrix}x \\ y \\ z \\ t\end{bmatrix}$ such that
$\displaystyle \begin{bmatrix}0 & 0 & 2 & 3 \\ 0 & -3 & -2 & -2\end{bmatrix}\begin{bmatrix}x \\ y \\ z \\ t\end{bmatrix}= \begin{bmatrix}0 \\ 0 \end{bmatrix}$
which means we must have 2z+ 3t= 0 and -3y- 2z- 2t= 0. Those two equations drop the dimension from 4 to 2. We can write z= (-3/2)t and then y= (-1/3)(2z+ 2t)= (-1/3)(-3t+ 2t)= (1/3)t. That is, y and z depend on t while x, since it does not appear in the equations can be anything. Use x and t as "free variables". Any vector in the kernel are of the form $\displaystyle \begin{bmatrix}x \\ (1/3)t \\ (-3/2)t \\ t\end{bmatrix}= x\begin{bmatrix}1 \\ 0 \\ 0 \\ 0\end{bmatrix}+ t\begin{bmatrix}0 \\ 1/3 \\ -3/2 \\ 1\end{bmatrix}$