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Lecture 2

Week 2 Lecture Notes.pdf

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Kathleen Smith

De▯nition: Let v ; 1 ; 2▯▯ ; v be vkctors in R . The span of these vectors is the set of all linear combinations of these vectors and is denoted by sp(v 1 v ;2▯▯▯ ; v )k sp(v 1 v 2 ▯▯▯ ; v )k= f▯ v 1 ▯1v + 2▯▯2+ ▯ v j ▯ ; ▯k▯ k▯ 2 1g. k De▯nition: Let v = [v 1v ;2▯▯▯ ;v ]nbe a vector in R . The norm p (or magnitude or length) of v is kvk = v + v + ▯▯▯ + v . 2 1 2 n Theorem (Properties of Norm) n For all vectors v; w 2 R and for all scalars ▯, we have (1) kvk ▯ 0 and kvk = 0 if and only if v = 0. (positivity) (2) k▯vk = j▯jkvk (homogeneity) (3) kv + wk ▯ kvk + kwk (triangle inequality) n De▯nition: A vector in R , with norm 1, is called a unit vector. n If v 6= 0 is a vector in R , a unit vector with the same direction as 1 v is v. kvk Note: The standard basis vectors are unit vectors and are sometimes called unit coordinate vectors. De▯nition: The
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