Rank(A) + nullity(A) = no. of cols of A (WHY?)

[nyxynyx]

Hello! I am confused over why rank(A) + nullity(A) = n = no. of columns of A, not no. of rows or something else.

My lecturer showed me something like a mxn matrix postmultiplied with a x-vector to get R^n, thus n = no. of cols. Makes sense when he was explaining but when i stepped out i realized that i didnt get it. Any help pls? Thanks!

 

[JasonRox]

You can probably find a proof somewhere online.

It's pretty quite intuitive after you get further into Linear Algebra and have become more comfortable with Nullspaces and such. I'll find a link with a proof.

 

[JasonRox]

I can't one what doesn't use linear transformations!

 

[nyxynyx]

My module hasn't reached transformations yet :(. Is there a explanation why its equal to no. of columns without talking about transformation?

 

[Hurkyl]

In my humble opinion, the rank-nullity theorem is not something you really want to "explain" -- it should be part of the foundation of your intuition for linear algebra. If you're looking for understanding, your best bet is probably to review previous exercises where you actually solved systems of equations and apply the rank-nullity theorem to describe the system.

e.g. you may have done an exercise solving Ax=b, where A is 3x3, and got a two-dimensional space of solutions. The rank-nullity theorem says that the rank of A is one -- so confirm that by computing the rank of A!

 

[JasonRox]

In my humble opinion, the rank-nullity theorem is not something you really want to "explain" -- it should be part of the foundation of your intuition for linear algebra. If you're looking for understanding, your best bet is probably to review previous exercises where you actually solved systems of equations and apply the rank-nullity theorem to describe the system.

e.g. you may have done an exercise solving Ax=b, where A is 3x3, and got a two-dimensional space of solutions. The rank-nullity theorem says that the rank of A is one -- so confirm that by computing the rank of A!

Exactly. It comes around. If it hasn't yet, continue solving systems.

 

[radou]

Once you learn something about linear operators and their matrix representation, it should become formally clear.

Edit: actually, you can investigate this fact by going into Gaussian elimination.