MAT211_Lecture14[1]

MAT211_Lecture14[1] - Denition MAT211 Lecture 14 Orthogonal...

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MAT211 Lecture 14 Orthogonal projections and orthogonal basis Orthogonality, length, unit vectors Orthonormal vectors: defnition and properties Orthogonal projections: defnition, Formula and properties. Orthogonal complements Pythagorean theorem, Cauchy inequality, angle between two vectors Defnition Two vectors u and v in R n are perpendicular or orthogonal iF u.v=0 The length oF a vector v in R n is ||v||= ! v.v A vector v in R n is called a unit vector iF ||v||=1 Example ±ind a unit vector in the line oF multiples oF (1,1,3) ±ind a vector oF length 2 orthogonal to (1,1,3) Defnition A vector v in R n is orthogonal to a subspace V oF R n iF it is orthogonal to all vectors in V Remark: IF (b 1 ,b 2 ,..,b m ) is a basis oF V, then v is orthogonal to V iF (and only iF) v is orthogonal to b 1 ,b 2 ,.. and b m . EXAMPLE Consider the subspace V oF R 3 span by (1,1,1) and (1,0,1). ±ind all the vectors orthogonal to V.
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MAT211_Lecture14[1] - Denition MAT211 Lecture 14 Orthogonal...

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