 Well the fundamental object of linear algebra is the matrix. So this leads naturally to the question. What is the matrix? And the matrix is just an extension of the vector concept So we can think about this as an n tuple of m tuples So I have some basic tuple and I'm going to take a set of them So for example vectors are tuples. So here is a vector and both of these are three tuples So I have two vectors and so I can take the two tuple of vectors v u and that's going to give us a matrix It's convenient not really necessary But it certainly makes it a lot easier to look at a lot easier to interpret To have one set of vectors written out horizontally and the other set written vertically And so for a variety of reasons mathematicians choose to write the base vectors vertically So here's my first vector v one three negative two We're going to set that up as a column and a second vector one negative one four And I'm going to set that up also as a column So now remember tuples do have an order to them So it does make a difference whether I put this one first or this one first and again I've chosen to put this one first this one second Also to keep the lines of text from running into each other We also as with tuples will throw a set of parentheses around to indicate that I now have a matrix Now there's a couple of terms that we use when we talk about Matrices and so we have the dimension of a matrix That's going to be n by m where n always refers to the number of rows the number of horizontal rows and m refers to the number of columns the number of vertical columns So that's how to mention if I have a matrix a the entries of that matrix are going to be usually lower case the same letter ij where ij is the value in the i-th row jth column Now if i and j are Numbers which they are because they're telling us for example third row fifth column The one problem with this particular notation is that if we write that those tend to look like a single number So we often see the notation a i comma j to express the entries of a matrix And one important idea given any sort of matrix a the entries a i i The entries where the i and i the row and column values are the same these form what's known as the main diagonal of the matrix So for example, let's take a matrix that looks like this and we want to describe the dimensions and find a couple of entries Including the main diagonal so the dimensions correspond to the number of rows and the number of columns So we count them so this matrix has one two three horizontal rows and one two three four vertical columns So this is a three by four matrix again remember row always goes first column always goes second Now again the entry a ij is going to be the value in the i-th row jth column So when I find this entry a three one, I'm looking at the entry in the third row first column So that's going to be one two third row first column that entry is going to be eight Likewise this entry a one comma three well here I've split up those That location so this is still first thing is the row second thing is a column This is the first row third column and so first row is this one third column over that gets to be the entry negative one so a one comma three is going to have value negative one a Three five is the one two third row one two three four fifth column entry except there isn't any And finally my main diagonal those are going to be the entries where the row and column number are the same So that's a one one a two two a three three so I can pick those off first row first column is three Second row second column is four third row third column is going to be negative one And there isn't going to be any entries after that