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Lecture 2: Geometry vs Linear Algebra Points-Vectors and Distance-Norm Shang-Hua Teng 2D Geometry: Points 2D Geometry: Cartesian Coordinates y (a,b) x 2D Linear Algebra: Vectors y (a,b) 0 x 2D Geometry and Linear Algebra • Points • Cartesian Coordinates • Vectors 2D Geometry: Distance 2D Geometry: Distance How to express distance algebraically using coordinates??? Algebra: Vector Operations • Vector Addition v1 w1 v1 w1 v and w then v w v w v w 2 2 2 2 • Scalar Multiplication 3v1 v1 3v and v 3 v v 2 2 Geometry of Vector Operations • Vector Addition: v + w v+w v w Geometry of Vector Operations • -v 2v v -v Linear Combination Linear combination of v and w {cv + d w : c, d are real numbers} Geometry of Vector Operations • Vector Subtraction: v - w v+w v v-w w Norm: Distance to the Origin • Norm of a vector: || v || v1 v2 2 2 v1 v v2 Distance of Between Two Points dist( v, w) || v w || v1 w1 v2 w2 2 2 v v-w w Dot-Product (Inner Product) and Norm v w v1w1 v2 w2 || v || v v Angle Between Two Vectors v w Polar Coordinate r v v (r cos , r sin ) r (cos , sin ) Dot Product: Angle and Length • Cosine Formula v rv (cos , sin ) and w rw (cos , sin ) v w rv rw cos cos sin sin rv rw cos( ) ||v||||w| | cos v w Perpendicular Vectors • v is perpendicular to w if and only if vw 0 Vector Inequalities • Triangle Inequality || v w || ||v|| ||w|| • Schwarz Inequality | v w | ||v||||w|| Proof: | v w | |v||||w|| | cos ||| v |||| w || 3D Points z y x 3D Vector z v (v1 , v2 , v3 ) y x Row and Column Representation v (v1 , v2 , v3 ) v1 v v2 v3 Algebra: Vector Operations • Vector Addition v1 w1 v1 w1 v v2 and w w2 then v w v2 w2 v3 w3 v3 w3 • Scalar Multiplication 3v1 v1 3v 3v2 and v v2 3v3 v3 Linear Combination • Linear combination of v (line) {cv : c is a real number} • Linear combination of v and w (plane) {cv + d w : c, d are real numbers} • Linear combination of u, v and w (3 Space) {bu +cv + d w : b, c, d are real numbers} Geometry of Linear Combination u v u Norm and Distance • Norm of a vector: z || v || v1 v2 v3 2 2 2 v (v1 , v2 , v3 ) y x • Distance dist( v, w) || v w || v1 w1 v2 w2 v3 w3 2 2 2 Dot-Product (Inner Product) and Norm v w v1w1 v2 w2 v3 w3 || v || v v v w ||v||||w| | cos Vector Inequalities • Triangle Inequality || v w || ||v|| ||w|| • Schwarz Inequality | v w | ||v||||w|| Proof: | v w | |v||||w|| | cos ||| v |||| w || Dimensions • One Dimensional Geometry • Two Dimensional Geometry • Three Dimensional Geometry • High Dimensional Geometry n-Dimensional Vectors and Points v (v1 , v2 ,, vn ) v1 v2 v' vn Transpose of vectors High Dimensional Geometry • Vector Addition and Scalar Multiplication n • Dot-product v w vk wk k 1 • Norm || v || v v n vk 2 k 1 • Cosine Formula v w ||v||||w| | cos High Dimensional Linear Combination • Linear combination of v1 (line) {c v1 : c is a real number} • Linear combination of v1 and v2 (plane) {c1 v1 + c2 v2 : c1 ,c2 are real numbers} • Linear combination of d vectors v1 , v2 ,…, vd (d Space) {c1v1 +c2v2+…+ cdvd : c1,c2 ,…,cd are real numbers} High Dimensional Algebra and Geometry • Triangle Inequality || v w || ||v|| ||w|| • Schwarz Inequality | v w | ||v||||w|| Basic Notations • Unit vector ||v||=1 • v/||v|| is a unit vector • Row times a column vector = dot product n v w vk wk v1 v2 k 1 w1 w vn 2 wn Basic Geometric Shapes: Circles (Spheres), Disks (Balls) x c x c r 2 x c x c r 2