Question

how do you find the nullspace and range?

Answer 1

I don't know how to find it out before seeing the problem. :)

Answer 2

Suppose you have a matrix \(A\).
Write \(A{\bf x}=0\), and solve this equation in a parametric form.
Say that with parameters \(t_i\) and vectors \({\bf v_i}\) you obtained the solution in a parametric vector form to be,

\({\bf x}=t_1{\bf v_1}+t_2{\bf v_1}+t_3{\bf v_3}+...+t_k{\bf v_k}\)
Then, your Null Space is a matrix formed by the columns of \({\bf v_i}\).
That is, Null(A)=\(\left[{\color{red}{\left.\begin{matrix}.. \\ ..\\ ..\\ ..\end{matrix}\right.}~~\color{green}{\left.\begin{matrix}.. \\ ..\\ ..\\ ..\end{matrix}\right.}}~~\color{blue }{\left.\begin{matrix}.. \\ ..\\ ..\\ ..\end{matrix}\right.}\right]\)

Answer 3

Where each ith column is the ith parametric vector in the solution for x.

Answer 4

(I'm kind of poor in writing it out ... (Haven't ever written a linear alg tutorial))

Answer 5

And the Range of A, is the span of the columns of A.
So if \(A=I_4\) for example, then (obviously) the range is \(\mathbb{R}^3\).

Answer 6

excuse me R^4.

Answer 7

Well, then, for any m\(\times\)m matrix, if your vectors are linearly independent, you will span \(\mathbb{R}^m\). (It doesn't have to be a square matrix though.)

Answer 8

this is kinda a push or a little kicker for the general question you expressed.
I hope this is somewhat helpful (:)

Answer 9

Well can you show me some examples. I also want to add on to this question. What is a nullity.

Answer 10

\[H:P_3\rightarrow \mathbb{R}^2 \]
given by \[ax^2+bx+c \rightarrow \left(\begin{matrix}a+b \\ a+c\end{matrix}\right)\]

Answer 11

@SolomonZelman

Answer 12

when a = -b and b =c, then you have the matrix is 0 matrix, right?
so, the nullity is all polynomial degree 2 \(ax^2 +bx +c| a= -b , b =c\)

Like that: \(4x^2-4x-4\) is a member of nullity.

Answer 13

This is hard. I know those what make it to 0 but how do you formally solve it step by step? I don't know what you did in the last step either