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The solutions to quiz #1 for the ece s-512 vector calculus course held in winter 2006. The quiz covers topics such as spanning a space, finding vector norms, orthogonality, and expressing a vector as a linear combination of two given vectors. Students are encouraged to use this document as a study aid for reviewing the concepts covered in the quiz.
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Winter 2006 Quiz # 1 ECE S-
You are given the following two vectors r 1 = [1 2]t^ and r 2 = [2 1]t. (a) Can you span the R^2 space using r 1 and r 2? Why? (b) Find || r 1 || and || r 2 ||. Are r 1 and r 2 orthogonal? Why? (c) If you are given the vector r = [3 1]t^ , can you express r as a linear combination of r 1 and r 2 , i.e., r = C 1 r 1 + C 2 r 2? Why? Find C 1 and C 2. Why aren’t C 1 and C 2 just the respective projections of r onto r 1 and r 2? (d) If we want to span the R^2 space using two orthonormal vectors φ 1 and φ 2 such that r 1 is in the direction of φ 1. Find φ 1 and φ 2 in terms of r 1 and r 2. (e) If we want to approximate r in part (c) by either r 1 or r 2 , which one would you use and why?