MATA23H3 Lecture Notes - Lecture 4: Diagonal Matrix, Triangular Matrix, Coefficient Matrix
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Mata23 - lecture 4 - inequalities, matrices, and linear systems. Inequalities: example 12 (continued): show that in r2 the nonzero vector (cid:126)n = [a, b] is perpendicular to the line ax + by + c = 0. (cid:126)u on the line ax + by + c = 0, (cid:126)u (cid:126)n = 0. Let (cid:126)u = (x2 x1, y2 y1) (x1, y1) and (x2, y2) are on the line. Ax1 + by1 + c = 0 (#1) ax2 + by2 + c = 0 (#2) #2 - #1: a(x2 x1) + b(y2 y1) = 0. [a, b] [x2 x1, y2 y1] = 0. (cid:126)n (cid:126)u = 0: example 13: let (cid:126)v (cid:126)w. prove that, ||(cid:126)v + (cid:126)w||2 = ||(cid:126)v||2 + || (cid:126)w||2, ||(cid:126)v (cid:126)w||2 = ||(cid:126)v||2 + || (cid:126)w||2. Prove ||(cid:126)v (cid:126)w||2 = ((cid:126)v (cid:126)w) ((cid:126)v (cid:126)w) ||(cid:126)v (cid:126)w||2 = ||(cid:126)v||2 || (cid:126)w||2: example 1: consider a linear system (cid:26) x1 2x2 = 1.
