Lecture2 - Sophie Chrysostomou Digitally signed by Sophie...

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Unformatted text preview: Sophie Chrysostomou Digitally signed by Sophie Chrysostomou DN: cn=Sophie Chrysostomou, o=UTSC, ou=Division of Computer and Mathematical Sciences, email=chrysostomou@utsc.utoronto.ca, c=CA Date: 2008.01.09 19:03:42 -05'00' Edited by Foxit Reader Copyright(C) by Foxit Software Company,2005-2007 For Evaluation Only. University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT A23 Lecture 2 Main ideas from lecture 1: 1. We defined algebraically and geometrically addition and subtraction between two vectors (BUT NOT BETWEEN TWO POINTS). 2. We defined algebraically and geometrically scalar multiplication of a vector ( BUT NOT OF A POINT). 3. We defined linear combination of vectors and the span of a collection of vectors. Problem: Find the vector x that starts at the point (1, 3) and ends at the point (4, −1). a = [1,3] b = [4,-1] x=b-a x = [4, -1] - [1, 3] x = [3, -4] Winter 2008 Homework : Find the vector b that starts at the point (x1 , x2 , · · · , xn ), and ends at the point (y1 , y2 , · · · , yn ). If v = [v1 , v2 , · · · , vn ] ∈ Rn and w = [w1 , w2 , · · · , wn ] ∈ Rn then we say v = w if vi = wi for all i = 1, 2, · · · , n, otherwise v = w. This is the only way we can compare two vectors. i.e. we cannot say that a vector v is ”less” (or ”greater”) than a vector w. However, we can compare their magnitudes. definition 0.1. The magnitude or norm of a vector u = [u1 , u2 , · · · , un ] ∈ Rn is denoted by u and is defined to be u= u 2 + u2 + · · · + u 2 1 2 n 1 Edited by Foxit Reader Copyright(C) by Foxit Software Company,2005-2007 For Evaluation Only. A vector with the magnitude of 1 is called a unit vector. Geometric Interpretation of Magnitude of Vectors. In R ||[a]|| = sqrt(a^2) = |a| In R2 : ||v|| = sqrt(a1^2+a2^2) = length of v In R3 ||w|| =sqrt(a1^2 +a2^2+a3^2)=length of w theorem 0.2. Properties of the Norm in Rn ∀ v, w ∈ Rn and for all scalars r, we have: 1. 2. 3. proof. v ≥ 0 and v = 0 ⇐⇒ v = 0 rv = |r| v v+w ≤ v + w ||rv||=||[rv1,rv2,rv3.....rvn]|| = sqrt() Positivity Homogeneity Triangle Inequality 2 example 0.3. Find the norm of u = [3, −1, 2, 5, 0], 3u, v = [1, 0, −2, 1, 4], 2v and 3u − 2v definition 0.4. Let v = [v1 , v2 , · · · , vn ] and w = [w1 , w2 , · · · , wn ] be n vectors in R . The dot product of Rn is defined to be the real number v · w = v1 w1 + v2 w2 + · · · + vn wn Claim: The angle between two nonzero vectors v and w in R2 is arccos v·w vw . definition 0.5. The angle between two nonzero vectors v and w in R2 is v·w arccos . vw 3 ...
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