I&C 6N. Computational Linear Algebra

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1 I&C 6N Computational Linear Algebra 1

2 Lecture 1: Scalars and Vectors What is a scalar? Computer representation of a scalar Scalar Equality Scalar Operations Addition and Multiplication What is a vector? Computer representation of a vector Size/Dimension of a vector Vector equality Scalar-Vector Operations Scaling of a vector Length/Magnitude of a vector Unit vector or Direction of a vector 2

3 What is a scalar? Scalar is simply a real number, such as b R (in general b ε C) Scalar can be negative or positive. In linear algebra, scalar is dimensionless, and does not represent any dimension. Example: 3, 4.2, -10, 2, π,, -, 0 3

4 What is NOT a scalar? Anything that is not a number such as text Later we will see that other objects like vector, matrices, etc. are also not scalars. 4

5 When are two scalars equal? Two scalars are equal when their sign and value are equal. This is the same equality as used in arithmetic. Example: 2.2 = = 2 / 0 = 2.5 5

6 What are the scalar operations? Any operation that can be done on real numbers, can be done on scalars. Examples: addition, subtraction, multiplication, division, etc , 2 3, 2 3, 0, , 2 6

7 What is a Vector? Vectors are the most fundamental entities in linear algebra. Vectors are sequences of scalars. Vectors are enclosed in brackets. Vectors representation is computer programming is the same as a 1-D array. Vectors can be represented as row or column vectors. Example: Column vector 1 0, Row Vector , 2 Note: 2 is a vector. 2 is a scalar. 7

8 Dimension of a vector Each scalar in a vector is called a component. Number of components of a vector is called the dimension or size of that vector. Example: Dimension of 1 0 is 2 Dimension of 0 3, - is 3 Dimension of 2 and [ 3.1] is 1. 8

9 What does a Vector represent? A Vector represents Magnitude (or length) and Direction. Direction 9

10 What is the graphical or geometric representation of a Vector? It is an arrow from the origin Length of the arrow is the magnitude and the direction is the direction of the vector. Magnitude = 5 Direction: 4 in dimension 1 3 in dimension 2 Origin Dimension 2 3 Dimension

11 When are two vectors equal? Two vectors are equal when their magnitude and direction is the same. This only happens when the two vector have the same dimension and their corresponding components are the same Example: 2 1 = 2 1 = 4/2 cos0 θ + sin 0 θ =

12 What is the point of application of a vector? Vectors are graphically represented by arrows. The origin point of an arrow is the point of application of a vector. Vector 4 3 from point 0 0 Vector 4 3 from point 0 1 The two vectors are equal Note that both the blue and 4 red vectors are, while the 3 coordinates of the start and end points of these two vectors are different. (0,1) (0,0) (4,4) (4,3) 12

13 Scalar-Vector Operation Vectors and scalars can be multiplied together Result is another vector of the same dimension as the input vector. By multiplying a vector and a scalar each component of the vector is multiplied by the scalar. ca = ac = , 0 2 =

14 Scaling a vector The magnitude/length of the vector does not change if the scalar is 1 or -1. The direction of the vector does not change if the scalar is positive. The direction of the vector becomes opposite, if the scalar is negative. Any vector of dimension N, when multiplied by a scalar 0, will result in a zero vector of dimension N (each component is 0). A zero vector has no direction, and its length is , 0 0 =

15 Geometric representation of scaling a vector Multiplying a scalar with a vector is like scaling that vector. Negative scalar changes the direction of the vector. Example: 2a a 0. 5a 1 2 a 15

16 Geometric representation of scaling a vector Multiplying a vector by 0 results in a vector with zero magnitude but does not change the direction. Example: 2a a 0. 5a 0a 1 2 a 16

17 How to compute the magnitude of a vector? (Ch. 6.1 P. 331) The magnitude or length of vector can be calculated by extending the Pythagoras formula into N dimension, and it is always a non-negative scalar. a, If a = a 0, the magnitude of a, or a = a 0, + a a 0 7 a 7 Example: = 5 17

18 Distance between Points (Ch. 6.1 P. 333) Given two points P, = (x,, y,, z, )and P 0 = x 0, y 0, z 0 x, The vector to P, from the origin is v, = y, z, the origin is v 0 = x 0 y 0 z 0 The vector from P, to P 0 is v 0 -v, = Distance between P, and P 0 is v 0 v, x 0 x, y 0 y, z 0 z, and vector to P 0 from 18

19 What is a Unit Vector and how to compute it(ch. 6.1 P. 332) A Unit Vector is a vector of length 1. Given any vector, its unit vector can be calculated. If a = Example: a = a, a 0 a 7, its unit vector a[ = a a 2 1 2, a[ = 2, 1 0 \ ] ^, \ ]0 \ 2 =, = 0 7, Direction of a vector is its unit vector. 19

20 For which vector, its unit vector cannot be computed? Only for zero vector, we cannot compute a unit vector The length of zero vector is zero. We cannot divide any number by zero. Magnitude (length) of a zero vector is 0. The zero vector does not have a direction, but is assumed a direction of the vector involved in the operation. For example 0.a=0. [Note: 0 is a scalar. 0 is a zero vector of the same dimension as vector a.] Although 0 doesn t have a direction, it is assumed to have the same direction as a for practical purposes. 20

21 How to represent a vector as a tuple of magnitude and unit vector? Any nonzero vector can be represented as a tuple of its magnitude and unit vector, such as a, a[. a = a a[ = a, a a Example: a = 4 3 a = 5 a[ = a[ a=5a[ 21

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