 In the previous video we learned how to compute the matrix vector product. That is if we have an m by n matrix And we multiply it by a vector from fn. We can get a vector in fm and so now consider the matrix equation ax equals b right here And so as we're talking on this matrix equation think of the following the vector b will be a specific fixed vector in fm so this will be given to you and likewise the the matrix a Which is an m by n matrix, which will also be fixed. This is given to you On the other hand the vector x which lives in fn. This is meant to be a variable This is what we're trying to solve for and so in section 2.2. This is the matrix equation We're concerned about ax equals b well given the way that we defined matrix vector multiplication you can write the So assuming a has it's column vectors a1 a2 Up to a n and so these are in fact vectors So if you have that a has column vectors a1 a2 up to a n then the Actually, well then it's or it's already right here then the matrix vector product is exactly this linear combination right here Where as we mentioned before the columns of a are just the vectors a1 up to a n and if the entries of x are x1 x2 Up to xn Expanding that product gives us this linear combination. So this matrix equation 2.2.1 is just equivalent to the vector equation We saw in the previous section and as we've learned this vector equation is equivalent to a system of linear equations Which is then represented by this augmented matrix Which the coefficient matrix is just the matrix a itself and then the augmented column is the vector b so the matrix equation ax equals b is equivalent to the linear system represented by the by the Augmented matrix a augment b and so we've defined matrix vector multiplication exactly so it Coincides with this notion of linear systems and so this just gives us a different way of writing the same thing So consider the linear system We have two equations three unknowns the first equation is x1 plus 3x2 minus 7x3 equals 5 The second equation is negative 2x2 plus 11x3 equals 3 We can rewrite this linear system in a lot of different ways We could write this linear system as an augmented matrix We get 1 3 negative 7 augment 5 We also get 0 negative 2 11 augment 3 And so we've used this augmented matrix as a way of encoding the system And then we start doing row operations to this augmented matrix to help us solve it On the other hand this linear system is equivalent to the vector equation x1 Times the vector 1 0 plus x2 times the vector 3 negative 2 plus x3 times the vector negative 7 11 And that equals 5 and 3 there where of course the the coefficients in this linear combination are unspecified variables These are the exact same variables in the linear system Then the vectors in play coincide with well the columns of this augmented matrix You have 1 and 0 1 and 0 you have 3 and negative 2 3 and negative 2 7 oops Negative 7 and 11 want to get a slurpee there right you're gonna get that one right there And then finally the 5 and 3 gives us this right here. So there is this These the column vectors in the linear combination are exactly just the column vectors of this matrix But then we've now added a way of Representing this linear combination more succinctly as a matrix product We can essentially think that the linear combination is an expanded form if we want the factored form We get the following matrix equation The matrix a which is 2 by 3 is given as 1 3 negative 7 0 negative 2 11 which are just the coefficients of The are just the coefficients in the linear system Which are of course the coefficients of it's the coefficient matrix of the augmented matrix You times those by this variable vector x1 x2 x3 This is just x right here and then on the right hand side we get the vector b So we can rewrite a linear system as a vector equation. We can write it as a matrix equation and the solution The solution to the matrix equation the solution to the vector equation The solution to the linear system are all the same those three different problems all have the same Solution set and we can solve each and every one of those linear systems by row reducing the associated augmented matrix, so Let me be clear the augmented matrix doesn't have a solution set because the augmented matrix is just an encoding of the systems But the linear system the vector equation the matrix equation These are actually problems that can be solved We solve them using the augmented matrix and the solution to one of those like the matrix equation the solution to That one is the same as the vector equation It's the same as the linear system. And so in fact we then see that b So this equation ax equals b here it's consistent It's consistent exactly when b is a linear combination of the column vectors because this matrix equation can be written as x1 a1 plus x2 a2 All the way up to x in an this equals b So this the the matrix equation being consistent is when this vector equation is consistent as we saw in the previous section This equation is consistent exactly when b is in the span Of these vectors here in this case would be the column vectors of the matrix a1 a2 up to an So solving the matrix equation is equivalent to solving All these other other equations we've seen so far in linear algebra