Linear Algebra Chapter 4 Vector Spaces 4 1
Linear Algebra Chapter 4 Vector Spaces
4. 1 The vector Space Rn Definition 1: …………………………………. The elements in Rn called …………. Addition and scalar multiplication: Definition 2: Let k scalar. be two elements of Rn. Addition scalar multiplication Ch 04_2
4. 1 The vector Space Rn Ex 1: Let Fined: be vectors in . Note: Ch 04_3
Theorem 4. 1 Properties of vectors Addition and scalar multiplication: Let u, v, and w be vectors in Rn and let c and d be scalars. 1) u + v = 5) c(u + v) = 2) u + (v + w) = 6) (c + d) u = 3) u + 0 = 7) c (d u) = 4) u + (–u) = 8) 1 u = Ex 2: Let u = (2, 5, – 3), v = ( – 4, 1, 9), w = (4, 0, 2) in R 3. Determine the linear combination 2 u – 3 v + w. Solution Ch 04_4
T h e n : Column Vectors Row vector: Column vector: and Ch 04_5
4. 2 Dot Product, Norm, Angle, and Distance Definition Let be two vectors in Rn. The ………………. of u and v is denoted ……. . and is defined by: Example 3 Find the dot product of u = (1, – 2, 4) and v = (3, 0, 2) Solution Ch 04_6
Properties of the Dot Product Let u, v, and w be vectors in Rn and let c be a scalar. Then 1. u. v = 2. (u + v). w = 3. cu. v = 4. u. u …… , and u. u = ……. u = …. . Ch 04_7
Norm of a Vector in Rn Definition The norm of a vector u = (u 1, …, un) in Rn is: …………………. . Definition A unit vector is a vector whose norm is …. . . (………) If v is a nonzero vector, then the vector ………. is a unit vector in the direction of v. This procedure of constructing a unit vector in the same direction as a given vector is called ……. ………. Ch 04_8
Example 4 Find the norm of each of the vectors u = (2, -1, 3) of R 3 and v = (3, 0, 1, 4) of R 4. Normalize the vector u. Solution ……………………………………………………………………. . Example 5 Show that the vector u=(1, 0) is a unit vector in R 2. Solution Ch 04_9
Angle between Vectors (in R n) Definition Let u and v be two nonzero vectors in Rn. The cosine of the angle between these vectors is Example 6 Determine the angle between the vectors u = (1, 0, 0) and v = (1, 0, 1) in R 3. Solution Ch 04_10
Orthogonal Vectors Definition Two nonzero vectors are ……………. . if the angle between them is a right angle. Theorem 4. 2 Two nonzero vectors u and v are orthogonal Example 7 Show that the vectors u=(2, – 3, 1) and v=(1, 2, 4) are orthogonal. Solution Ch 04_11
Note (1, 0), (0, 1) are orthogonal unit vectors in R 2. ………. , ………. are orthogonal unit vectors in R 3. ………. , … , ………. , are orthogonal unit vectors in R n. Distance between Points Let be two points in Rn. The …………… between x and y is denoted …. . and is defined by: Example 8. Determine the distance between x = (1, – 2 , 3, 0) and y = (4, 0, – 3, 5) in R 4. Solution Ch 04_12
Properties of Norm: Properties of Distance: Ch 04_13
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