 So yesterday, I showed you very briefly this column view. Now, I didn't mention really, I think the word column view, but there's a way not to see matrices as rows, but as columns. There's really two views on the matrices. We have row operations. That's really just seeing these things as rows, but much more interesting is the column view. So here I have two equations and two are known. Instead of x and y, I've made them x sub 1 and x sub 2 just so that we don't get that confusion with the x and y's again. And I can have my augmented matrix. Let's just have that first. My augmented matrix. And that would be a 2, a 1 and a 7, and a 1 and negative 1 and a 2. You can even do that. And if we do gas Jordan elimination, I'm going to get to 1, 0, 0, 1, and I'm going to get to a 3 and a 1. Just to indicate that x sub 1 must be 3 and x sub 2 must be 1. So that's one way of looking at it. I could also have done the following. I could just have my matrix of coefficients call that A and that would be a 2 and a 1 and a 1 and a negative 1. And I would have my matrix, my column vector x, I should say. And that would be x sub 1 and x sub 2. And I showed you that if I have A times x, my matrix times a vector, that's why we looked at that. We're going to get this solution vector and my solution vector B was going to equal 7 and 2. And I showed you if I were to write this A and x like that and I did that multiplication, I was indeed going to end up with that and that has got to equal B on that side. And if we could get A inverse, I would have A inverse A and x and that would equal A inverse B. In other words, x was going to equal A inverse B. So if I could get that, that's what I showed you. What I'm really interested in here is this idea of looking at it in column form and we'll look at that because it's much, much more interesting. Let's have a look at that. What we are really seeing here is the following. I have this column vector 2 and 1 and I have another column vector there which is a 1 and a negative 1. And what I'm trying to get to is this column vector 7 and 2. This is what I'm looking at here and I'm saying I'm looking for some scalar multiple of this column vector and I want to add to that some scalar multiple of the second vector to get this column vector. Is that so? Yes it is because I want 2 times x1 plus x2 to give me 7 and I want x1 minus x2 to give me 2. So everything holds. But I want to see all of this as a column vector and all of this as a column vector because look at this. If I see this in a two-dimensional plane as a column vector there's 2, 1 which is this there and there is 1, minus 1. You'll have to watch this on a large screen to see. There's 1, negative 1. And my solution vector is this one 7, 2. And I'm asking how many times this vector added to how many times this vector will give me that vector. Can I get to there by a linear combination of these two vectors? So if I have a linear combination means I've got to scale this vector along this line and this little vector I've got to scale along this line so that eventually I get to 2 of them and if I add so I might get to there and this one I might get to there or I might multiply by negative and get to there and get to there and if I add these two can I get to that vector? That is the column view. I hope you see this column view of a vector. I want to know how many times this column vector how many times that one plus how many times this one do I have to get to get to that one. And these coefficients that we have there, 2 and 1 those are vectors in this space that I have here, this two-dimensional space. And when we look at it and we see that x1 equals 3 and x2 equals 1, look at what happens. If I take this vector and I multiply it by 3 it means each component is multiplied by 3 and so I get this vector. This vector here becomes 6 and a 2 and if I multiply this by 1 I'm left with a 1 and a minus 1 so I'm saying 3 times this vector plus 1 times that vector and if I do that I land up at 6 and I land up at 3 which is right there and that is along the line of this vector. This vector has been multiplied by a scalar which just stretches it in this line and that one is just itself which is just 1 minus 1 which is right there and there I get to 7 and 2. So that is a linear combination of these column vectors to give me a third vector. And now you can see that these two vectors that I have here a linear combination of them can get me anywhere in the plane. Anywhere in the plane I can get to by linear combination of just these two and we say that these two vectors span this whole two-dimensional space and you can also see why now if my second vector was along this line if my second vector was along this line as well so say imagine it was just there or it was there no linear combination of them would get me anywhere off of that line so these two would not span two-dimensional space and we'll see that if we have linear combinations of them then we can't get to anything other than a combination of this space. So this illustration that I'm trying to make is that we see these matrices as not as or these systems of linear equations not as rows in a matrix not as an augmented matrix not as a matrix of coefficients but as this linear combination of column vectors to give me another column vector in the same space it's two rows so it's two-dimensional space two-dimensional space two-dimensional space I'm here in flat two-dimensional space and that is much more important and a much more interesting view of these systems of linear equations and of matrices is to see them as these linear combinations of column vectors the column view of a matrix.