1. Chapter 4 Vector Spaces Linear Algebra

2. Ch04_2 Definition 1: ……………………………………………………………………. The elements in R n called …………. 4.1 The vector Space R n Addition and scalar multiplication: Definition 2: Let be two elements of R n. k scalar. Addition scalar multiplication

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

4. Ch04_4 Theorem 4.1 Let u, v, and w be vectors in R n and let c and d be scalars. 1)u + v = 2)u + (v + w) = 3)u + 0 = 4)u + (–u) = Properties of vectors Addition and scalar multiplication: 5) c(u + v) = 6) (c + d) u = 7) c (d u) = 8) 1u = Let u = (2, 5, –3), v = ( –4, 1, 9), w = (4, 0, 2) in R 3. Determine the linear combination 2u – 3v + w. Ex2: Solution

5. Ch04_5 Column Vectors Then: and Row vector: Column vector:

6. Ch04_6 4.2 Dot Product, Norm, Angle, and Distance Definition Let be two vectors in R n. 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

7. Ch04_7 Properties of the Dot Product Let u, v, and w be vectors in R n and let c be a scalar. Then 1.u.v = 2.(u + v).w = 3. cu.v = 4.u.u ……, and u.u = ……. u = …..

8. Ch04_8 Norm of a Vector in R n Definition The norm of a vector u = (u 1, …, u n ) in R n 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 …….………..…….

9. Ch04_9 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 4 …………………………………………………………………….. Show that the vector u=(1, 0) is a unit vector in R 2. Example 5 Solution

10. Ch04_10 Definition Let u and v be two nonzero vectors in R n. 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 Angle between Vectors (in R n )

11. Ch04_11 Definition Two nonzero vectors are …………….. if the angle between them is a right angle. Two nonzero vectors u and v are orthogonal Theorem 4.2 Orthogonal Vectors Solution Example 7 Show that the vectors u=(2, –3, 1) and v=(1, 2, 4) are orthogonal.

12. Ch04_12 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 R n. 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

13. Ch04_13 Properties of Norm:Properties of Distance: