 One of the important activities of mathematics is taking something that quote everybody knows about and Examining it from every possible angle in the hopes that we'll find out even more than what everybody knows So let's take a deeper look at coordinates So let's start off with our set of vectors and let's consider any linear combination of this set of vectors and So notice that this linear combination is completely specified by the n Coefficient a1 through an so I can take the n-tuple a1 through an and call these the Coordinates of my vector x with respect to my set of vectors V Now at this point we haven't committed ourselves to what this vector set V looks like but if V is a basis for some vector space V then the Coordinates of this vector x will be unique and Conversely and this is something you should be able to prove Since it's possible for there to be more than one basis for a given vector space suppose I have two different basis sets for our vector space We might consider the following problem to convert the coordinates of x with respect to one basis Into coordinates with respect to the other basis So in order to solve this problem what we're going to need to do is we'll need to Set up and solve a system of linear equations so if the coordinates of x with respect to P are a1 through an and b1 through bn with respect to q then because they are supposed to be coordinates for the same point I know that the linear combination with the coefficients a1 through an has to be equal to the linear combinations b1 through bn We can view this in matrix form on the left hand side we have the matrix consisting of the columns p1 through pn and The column vector of the coordinates a1 through an on the right hand side We have the matrix consisting of the columns q1 through qn and the column vector corresponding to the coordinates with respect to q and Purely for notational convenience will let p be the matrix whose columns are the basis p and q be the matrix whose columns are the basis q and that allows us to write our equation as Matrix p times the column vector of the coordinates equals matrix q times the column vector of the coordinates Now by assumption, we know both p and q because we have the basis p and the basis q We also know all of the a i's because those are the Coordinates of our vector with respect to p so that says that we can solve for the b i's by multiplying by the multiplicative inverse of q Provided that this multiplicative inverse exists more generally. This is the left inverse of q So let's see how that works in practice So here I'll have a basis for a two-dimensional vector space I'll have a different basis for the same vector space and suppose the coordinates of some Vector x with respect to our first basis are three one Let's see if we can find the coordinates with respect to our second basis So let's go ahead and try and set that up if we have the coordinates of x with respect to u are u1 u2 then we know that the Product three five four seven times three one is going to be the product one one four three times u1 u2 And again, it'll help to remember what this really means this is a linear combination that consists of taking three of the vector three four and one of the vector five seven and Those correspond to the coordinates of x with respect to our first vector space v what we want to do is we want to find the linear combination of 1 4 and 1 3 that give us the same vector So what I can do to solve for this vector u1 u2 I can multiply on the left by the inverse of 1 1 4 3 and We can find that in any number of ways We find the inverse and we multiply on the left by it and get u1 u2 as a product of three matrices Which ends up being negative twenty three thirty seven and because there's a lot of places where the arithmetic may have gone wrong Let's verify this so what we're claiming is that the linear combination three of three four and one of five seven is Going to be the same as negative twenty three of one four and thirty seven of one three So let's check it out three of three four plus one of five seven is fourteen nineteen and negative twenty three of one four and thirty seven of one three is Also fourteen nineteen and so these are the coordinates with respect to our vector u