33A Practice Midterm Question 9 (Explanation and Solution)
DISCLAIMER: To do this problem you need to know 1) How to invert a 3x3 matrix and 2) How to multiply matrices, I will not be showing my work on how I did that, as that would take literally forever. That being said we are given: T(from beta to beta) = 3 1 1 2 5 2 5 6 3 We are also given our friend's basis vectors, these three vectors we will combine into a matrix that we'll give an arbitrary name "Q". Another name for Q is the Identity matrix (from gamma to beta). Q / [I] (from gamma to beta) = 1 2 2 0 3 1 2 2 0 If you remember from class, our homie David gave us the formula: Q^-1 * [T](from beta to beta) * Q = [T](from gamma to gamma) We are looking for [T](from gamma to gamma), so we'll be using this formula. For this formula, the first thing we need to find is the inverse of Q. Another name for the inverse of Q is the Identity matrix (from beta to gamma) Q^-1 / [I] (from beta to gamma) = 1/5 -2/5 2/5 -1/5 2/5 1/10 3/5 -1/5 -3/10 Now, the first step in the formula is to multiply the inverse of Q (Q^-1) by the linear transformation T ([T] from beta to beta) that we were given. Q^-1 * [T](from beta to beta) = 9/5 3/5 3/5 7/10 12/5 9/10 -1/10 -11/5 -7/10 And the final step to find our answer ([T] from gamma to gamma) is to multiply the matrix we just solved for (Q^-1 * [T] from beta to beta) with our friend's basis Q. Q^-1 * [T](from beta to beta) * Q = 3 33/5 21/5 5/2 52/5 19/5 -3/2 -41/5 -12/5 There's our final answer.
Mitch_Grove: May 5, 2015, 4:10 p.m.
Formatting is difficult on this website: http://imgur.com/cqxnG37 here's a link to the problem worked out (I used "S" as the name of my friend's basis instead of Q, but it doesn't matter)
danielkim: May 5, 2015, 5:15 p.m.
Since it's from our basis (beta) to our friend's basis (gamma), I think the identity matrix goes from beta to gamma. https://www.camscanner.com/share/dtwLx/0/w105s0bnkuztn
brian_li: May 6, 2015, 1:43 a.m.
@danielkim you are right. I think all of the math is correct, just the formula is incorrect
aleksanderjanczewski: May 6, 2015, 2:09 a.m.
@Mitch_Grove: I guess you might have made some mistake in multiplying matrices. Check out the link: http://www.wolframalpha.com/input/?i=%7B%7B0.2%2C-0.4%2C0.4%7D%2C%7B-0.2%2C-0.6%2C0.1%7D%2C%7B0.6%2C-0.2%2C-0.3%7D%7D*%7B%7B3%2C1%2C1%7D%2C%7B2%2C5%2C2%7D%2C%7B5%2C6%2C3%7D%7D*%7B%7B1%2C2%2C2%7D%2C%7B0%2C3%2C1%7D%2C%7B2%2C2%2C0%7D%7D
brian_li: May 6, 2015, 2:21 a.m.
Actually, we are both wrong @danielkim. It is confusing because the wording of the problem is a little bit tricky. When it says "your friend's FIRST basis is written", it means that your friend's (1,0,0) is your (1,0,2). Thus, it IS gamma to beta.