What does the rank of a matrix means and how can we find it for a 2*2 and 3*3 matrix in an easier way?
The rank of a matrix represents the dimension of the image under the linear transformation it represents. The rank is the number of pivot positions in the matrix. Alternatively, you can take various combinations of kxk minors, and the rank of the matrix is the largest kxk minor which has nonzero determinant. This is a very quick method to determine rank in small matrices.
Err.... Gamma, you kind of lost me there. lol.
Actually, I understood the bit about the pivots (which was pretty simple, I admit)- but what do you mean by, "The rank of a matrix represents the dimension of the image under the linear transformation it represents"? You mean, that, if you've got a linear system, say in $\displaystyle \mathbb{R}^4$ and the solution set is a plane (that is 2D and therefore 2 pivots) than the rank is 2? What do you mean by image? What is the map? Actually, I should sit down and learn this...
Okay, so a matrix A is a representation of a linear map right, call the map L? We are talking about a function $\displaystyle L:V\rightarrow W$ where V and W are vector spaces. The rank of the matrix A is the same as the rank of the map L. They are two ways of talking about the same thing.
For a matrix you find the rank by counting the pivot positions. For a map you see what the dimension of $\displaystyle L(V) \subset W$ is, whatever the dimension is, is the rank of the map. If A is the representation of L then these are the same.
A similar concept comes with the null space. You can see what the dimension of Ax=0 is, or you can ask what is the rank of the kernel of the map. This is the dimension of the vector space consisting of all the vectors that get sent to 0. These two are the same.
But on top of this you have the rank - nullity theorem which states that the dimension of V is the sum of the rank plus the nullity. Which makes sense in terms of the matrix because a column is either a pivot column or its not. The ones that are not count towards the null space and the ones that are count towards the rank.
Thanks Gamma. That is pretty damn cool- tbh. we haven't talked about matrices as linear maps, but it makes sense- the different column vectors map onto a vector sub-space. And the sub-space has some dimension between the zero vector and the n-dimensions of the vector space itself.
I'm finding one of the most difficult things about Linear Algebra is simply organizing the amount of information, so that it's coherent. I mean ,the inter-change between linear equation and matrix terminology isn't always easy to follow.
yeah, it definitely takes some getting used to, but it is a really convenient way to deal with linear mappings. Especially when you are composing them since you can just do matrix multiplication. Plus the determinant is really powerful. If you continue in mathematics you will encounter this sort of thing many more times.