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Chapter 4

Linear Algebra. Chapter 4. Vector Spaces. 4.1 The vector Space R n. Definition 1: ……………………………………………………………………. The elements in R n called …………. Addition and scalar multiplication:. Definition 2: Let be two elements of R n .

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Chapter 4

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  1. Linear Algebra Chapter 4 Vector Spaces

  2. 4.1 The vector Space Rn Definition 1: ……………………………………………………………………. The elements in Rn called …………. Addition and scalar multiplication: Definition 2: Let be two elements of Rn. k scalar. Addition scalar multiplication

  3. 4.1 The vector Space Rn Ex 1: Let be vectors in . Fined: Note: Ch04_3

  4. 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. • u + v = • u + (v + w) = • u + 0 = • u + (–u) = 5) c(u + v) = 6) (c + d) u = 7) c (d u) = 8) 1u = Ex2: Let u = (2, 5, –3), v = ( –4, 1, 9), w = (4, 0, 2) in R3. Determine the linear combination 2u – 3v + w. Solution Ch04_4

  5. Row vector: Column vector: Column Vectors Then: and

  6. Example 3 Find the dot product of u = (1, –2, 4) and v = (3, 0, 2) 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: Solution

  7. Properties of the Dot Product • Let u, v, and w be vectors in Rn and let c be a scalar. Then • u.v = • (u + v).w = • cu.v = • u.u…… , and u.u = ……. u = …..

  8. Norm of a Vector in Rn Definition The norm of a vector u = (u1, …, un) in Rnis: …………………………………….. 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 …….………..…….

  9. Example 4 Find the norm of each of the vectors u = (2, -1, 3) of R3 and v = (3, 0, 1, 4) of R4. Normalize the vector u. Solution …………………………………………………………………….. …………………………………………………………………….. …………………………………………………………………….. …………………………………………………………………….. Example 5 Show that the vector u=(1, 0) is a unit vector in R2. Solution

  10. Example 6 Determine the angle between the vectors u = (1, 0, 0) and v = (1, 0, 1) in R3. Angle between Vectors (in Rn) Definition Let u and v be two nonzero vectors in Rn. The cosine of the angle between these vectors is Solution

  11. Example 7 Show that the vectors u=(2, –3, 1) and v=(1, 2, 4) are orthogonal. Theorem 4.2 Two nonzero vectors u and v are orthogonal Orthogonal Vectors Definition Two nonzero vectors are …………….. if the angle between them is a right angle . Solution

  12. Example 8. Determine the distance between x = (1,–2 , 3, 0) and y = (4, 0, –3, 5) in R4. Note • (1, 0), (0,1) are orthogonalunit vectors in R2. • ………., ………., ………. are orthogonal unit vectors in R3. • ………., ………., … , ……….,are orthogonal unit vectors in Rn. Distance between Points Let be two points in Rn. The ……………between x and y is denoted ….... and is defined by: Solution

  13. Properties of Norm: Properties of Distance: Ch04_13

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