MATH 1ZA3 Lecture 22: Lecture 22

Prove Theorem i.e. prove that trace(AB) = trace(BA). Let A and B be matrices of size m times n and n times m respectively (so the both products AB and BA are well defined). Then trace(AB) = trace(BA) We leave the proof of this theorem as an exercise, see Problem below. There are essentially two ways of proving this theorem. One is to compute the diagonal entries of AB and of DA and compare their sums. This method requires some proficiency in manipulating sums in notation. If you sue not comfortable with algebraic manipulations, there is another way. We can consider two linear transformations, T and T1, acting from Mnxm to R = R1 defined by T(X) = trace(AX) , T1(X) = trace(XA) To prove the theorem it is sufficient to show t hat T = T1 the equality for X = B gives the theorem. Since a linear transformation is completely defined by its values on a generating system, we need Just to check the equality on some simple matrices. for example on matrices Xj, k, which has all entries 0 except the entry 1 in the intersection of jth column and kth row.

Here are two famous general formulas: and For example, 1 + 2 + 3 = 6 = 4 3/2. Works! What about Let's see whether we can make an educated guess. Expanding the right sides of our first two formulas gives and So, the formula for the first sum is of the form an2 + bn + c and the formula for the second sum is of the form an3 + bn2 + cn + d (with just the right choices of coefficients). So, maybe, the next formula is of the form

Example: Follow the steps to calculate the area between f(x) Bx and the x-axis, on the interval 12, 18 using Riemann sums la) Sketch the area that we want to find many rectangles do we need, to make sure we get the true área Inot an approximation)? lot 保 te) Divide the interval [2, 18] into N equal subdihisions of length Ax. (d) if we use岝to denote the nght-hand endpoint of the kih rectangle, what is xǐ in terms of k and N? lel Simplify Given: Atl [f) Areas Note: for this particular shape, we would work out the true area by finding the area of the big triangle, and subtracting the area of the smaller triangle. We get the same answer