Question

find a4x4 matrix A for which null space = col space
Please, help

Answer 1

Well, do you remember wha the null space and column space are?

Answer 2

If I recall, this is going to have something to do with it being orthagonal because then the null space and the null space of the transpose are the same and the column space is basically the row space of the transpose.

Answer 3

I have a 2x2 matrix satisfy it, but cannot find a 4x4

Answer 4

\[\left[\begin{matrix}1&1\\-1&-1\end{matrix}\right]\] colspace is \[\left[\begin{matrix}x\\-x\end{matrix}\right]\] the same with its null space.

Answer 5

isn't the result of a null space the 0 vector because Ax = 0. it can only have the trivial solutions right

Answer 6

yes

Answer 7

@galacticwavesXX no. It is \(A\vec{x}=\vec{0}\), but that is not always only trivial. For example, go to this site:

http://www.math.uh.edu/~jmorgan/Matrix/

Put in:
\(\left[\begin{matrix}1&1\\-1&-1\end{matrix}\right]\)
Solve for the null space.

Then put in:
\(\left[\begin{matrix}-1&0\\1&2\end{matrix}\right]\)
Solve for the null space.

You will find that the first is not trivial but the second is.

Answer 8

Yeah i forgot there are cases when it's not the trivial solution. We just worked on problems that had the null space as the empty set. i jumped into conclusions too quickly

Answer 9

The funny part is that the trivial solution is \((ax+by+cz=0|a=0,b=0,c=0)\) and they don't let you call it trivial until you have a graduate degree. LOL

Answer 10

we never learned that the equation was in 3D. our professor teaches it too us in a different way

Answer 11

Well, not the, but a trivial solution. There are an infinite number of them.

Answer 12

ohhh maybe we'll get there, we are busy proving a lot of theorems. Right now we are in the process of proving the Rank Theorem. That should be next class

Answer 13

The more technical description for it would be for any \(c_1x_1+c_2x_2+\ldots+c_nx_n=0\) where all constants c are 0 is the trivial solution. Because there are infinite possibilities for \(x_n\) there are infinite trivial solutions. They are simply trivial because it is anything and everything being multiplied by 0 adding up to 0. So the additive and multiplicative identities of 0 on steroids.