1. ENGG2013 Unit 4 Checking out at sushi bar Jan, 2011.

2. Row and Column Vectors column vector row vector n-dimensional column vector kshumENGG20132 n A list of n numbers written vertically Convention: a vector is by default a column vector in ENGG2013. Convention: The components in a vector are sometime called “scalar”.

3. Illustration kshumENGG20133 x y (0,0) (5,3) 5 3 (4,4) (1,-1)

4. Notation using column vector kshumENGG20134 x y 5 3

5. Notation from Physics kshumENGG20135 x y zhas the same meaning as

6. Interpretation of vector (I) kshumENGG20136 y (5,3) 5 3 location or

7. Interpretation of vector (2) kshumENGG20137 y (0,0) (5,3) 5 3 an arrow from the origin or

8. Interpretation of vector (3) kshumENGG20138 y (0,0) (5,3) 5 3 Any arrow in the same direction with the same length or

9. Mathematical Notation The set of all 2-D vectors with real numbers as components is denoted by The set of all 3-D vectors with real numbers as components is denoted by The set of all n-D vectors with real numbers as components: kshumENGG20139

10. Equality for vectors... … is just equality in each component Examples kshumENGG201310

11. Vector addition … … is just component-wise addition kshumENGG201311 But has no meaning.

12. Scalar Multiplication Multiply each component by the scalar constant. Interpreted as lengthening, or shortening the vector, but keeping the same direction. kshumENGG201312 x y z

13. Dot product A.k.a. scalar product, or inner product. For 2-D vector, It measures the “angle” between two vectors. – The dot product of two vectors is zero if the two vectors are perpendicular kshumENGG201313

14. Dot product in general For n-dimensional vectors in general, we define the dot product as Example kshumENGG201314 Two n-dim vectors are said to be perpendicular, or orthogonal, if their dot product is equal to 0.

15. Simple properties kshumENGG201315 For any two vectors u, v and w of the same dimension, and constants c and d.

16. Matrix-vector multiplication Given an m  n matrix A, and an n-dimensional vector x, the product of A and x is an m-dimensional vector defined as kshumENGG201316 For double subscripts, the first subscript is the row index and the second is the column index

17. Just compute dot products m times kshumENGG201317 Dot product of the first row in the matrix and the column vector Dot product of the second row in the matrix and the column vector Dot product of the last row in the matrix and the column vector

18. Example kshumENGG201318 is un-defined. 2x4 3x1 2x4 4x1 2x1

19. Checking out in a sushi bar Four prices – Red plate: $10 – Blue plate: $20 – Green plate: $35 – Pink plate: $50 kshumENGG201319 Customer A Customer B RBGP

20. Nutrition problem kshumENGG201320 In matrix notation: Amounts of food A, B, C and D Requirements of the four nutrients (protein) (carbohydrate) (vitamin A) (vitamin C)

21. Vector equation Equation involving vectors. E.g. Find a and b such that kshumENGG201321

22. The nutrition problem as vector equation Just another way to write the same thing. kshumENGG201322

23. Four different perspectives kshumENGG201323 A system of linear equations Ax = b Augmented matrix Vector equation