MATH 22A: LINEAR ALGEBRA Chapter 1

Transcription

1 MATH 22A: LINEAR ALGEBRA Chapter 1 Steffen Borgwardt, UC Davis original version of these slides: Jesús De Loera, UC Davis January 10, 2015

2 Vectors and Matrices ( ).

3 CHAPTER 1 Vectors and Linear Combinations 1.1

4 We can think of VECTORS as ordered n-tuples of real numbers. Each entry is a component. We often write them vertically! They are points in R n. 1 U = 12 U T = ( ) 7 Vectors can be added Get a vector again!! Vector can be multiplied by a number Get a vector again!! Definition For a finite set of vectors v 1, v 2,... v k, and numbers λ 1, λ 2,..., λ k a linear combination is the vector obtained as λ 1 v 1 + λ 2 v λ k v k The result is obtained operating one component at a time!! = 7 1

5 We can think of VECTORS as ordered n-tuples of real numbers. Each entry is a component. We often write them vertically! They are points in R n. 1 U = 12 U T = ( ) 7 Vectors can be added Get a vector again!! Vector can be multiplied by a number Get a vector again!! Definition For a finite set of vectors v 1, v 2,... v k, and numbers λ 1, λ 2,..., λ k a linear combination is the vector obtained as λ 1 v 1 + λ 2 v λ k v k The result is obtained operating one component at a time!! = =

6 We can think of VECTORS as ordered n-tuples of real numbers. Each entry is a component. We often write them vertically! They are points in R n. 1 U = 12 U T = ( ) 7 Vectors can be added Get a vector again!! Vector can be multiplied by a number Get a vector again!! Definition For a finite set of vectors v 1, v 2,... v k, and numbers λ 1, λ 2,..., λ k a linear combination is the vector obtained as λ 1 v 1 + λ 2 v λ k v k The result is obtained operating one component at a time!! = =

7 Addition of vectors is a commutative operation: A + B = B + A. There is a geometric interpretation of adding two vectors: What is A B in the picture?

8 Addition of vectors is a commutative operation: A + B = B + A. There is a geometric interpretation of adding two vectors: What is A B in the picture?

9 Addition of vectors is a commutative operation: A + B = B + A. There is a geometric interpretation of adding two vectors: What is A B in the picture?

10 QUESTION What happens if we take ONE vector A and compute ALL its linear combinations λa?? What is the picture?

11 QUESTION What happens if we take ONE vector A and compute ALL its linear combinations λa?? What is the picture? QUESTION Take two vectors A, B in the plane, compute all its linear combinations λa + µb with coefficients 0 λ, µ 1. What is the resulting picture?

12 QUESTION What happens if we take ONE vector A and compute ALL its linear combinations λa?? What is the picture? QUESTION Take two vectors A, B in the plane, compute all its linear combinations λa + µb with coefficients 0 λ, µ 1. What is the resulting picture? QUESTION Think of the 12 vectors that go from the center of a clock to the hours 1:00, 2:00,..., 12:00. What is the sum of these vectors?

13 QUESTION What happens if we take ONE vector A and compute ALL its linear combinations λa?? What is the picture? QUESTION Take two vectors A, B in the plane, compute all its linear combinations λa + µb with coefficients 0 λ, µ 1. What is the resulting picture? QUESTION Think of the 12 vectors that go from the center of a clock to the hours 1:00, 2:00,..., 12:00. What is the sum of these vectors? QUESTION Take two vectors A, B in the plane and consider all their linear combinations λa + µb with coefficients λ, µ INTEGER numbers. What do these linear combinations look like?

14 QUESTION What happens if we take ONE vector A and compute ALL its linear combinations λa?? What is the picture? QUESTION Take two vectors A, B in the plane, compute all its linear combinations λa + µb with coefficients 0 λ, µ 1. What is the resulting picture? QUESTION Think of the 12 vectors that go from the center of a clock to the hours 1:00, 2:00,..., 12:00. What is the sum of these vectors? QUESTION Take two vectors A, B in the plane and consider all their linear combinations λa + µb with coefficients λ, µ INTEGER numbers. What do these linear combinations look like? QUESTION Take two vectors A, B in the plane and consider all their linear combinations λa + µb with coefficients λ, µ any real numbers? What is the end result?

15 QUESTION Take three vectors A, B, C in (3-dimensional) space and consider all their linear combinations λa + µb + γc with coefficients 0 λ, µ, γ 1, what is the picture?

16 QUESTION Take three vectors A, B, C in (3-dimensional) space and consider all their linear combinations λa + µb + γc with coefficients 0 λ, µ, γ 1, what is the picture? QUESTION Take three vectors A, B, C in (3-dimensional) space and consider all their linear combinations λa + µb + γc with coefficients λ, µ, γ any real number, what is the picture?

17 QUESTION Take three vectors A, B, C in (3-dimensional) space and consider all their linear combinations λa + µb + γc with coefficients 0 λ, µ, γ 1, what is the picture? QUESTION Take three vectors A, B, C in (3-dimensional) space and consider all their linear combinations λa + µb + γc with coefficients λ, µ, γ any real number, what is the picture? QUESTION What do all linear combinations of the vectors 1 1 1, 0 in (3-dimensional) space generate? 1 0

18 QUESTION Take three vectors A, B, C in (3-dimensional) space and consider all their linear combinations λa + µb + γc with coefficients 0 λ, µ, γ 1, what is the picture? QUESTION Take three vectors A, B, C in (3-dimensional) space and consider all their linear combinations λa + µb + γc with coefficients λ, µ, γ any real number, what is the picture? QUESTION What do all linear combinations of the vectors 1 1 1, 0 in (3-dimensional) space generate? QUESTION What happens if you add the vector 1? 1 How does your answer change?

19 LENGTHS, ANGLES, DOT PRODUCTS (1.2) Wish: to measure the length of a vector, or the angle between two vectors

20 LENGTHS, ANGLES, DOT PRODUCTS (1.2) Wish: to measure the length of a vector, or the angle between two vectors Definition The dot product of two vectors p 1 q 1 p 2 p =., q = q 2. equals p n q n p q = p 1 q 1 + p 2 q p n q n

21 LENGTHS, ANGLES, DOT PRODUCTS (1.2) Wish: to measure the length of a vector, or the angle between two vectors Definition The dot product of two vectors p 1 q 1 p 2 p =., q = q 2. equals p n q n p q = p 1 q 1 + p 2 q p n q n = q p

22 LENGTHS, ANGLES, DOT PRODUCTS (1.2) Wish: to measure the length of a vector, or the angle between two vectors Definition The dot product of two vectors p 1 q 1 p 2 p =., q = q 2. equals p n q n p q = p 1 q 1 + p 2 q p n q n = q p Definition The length of a vector is defined as v = v v

23 Proposition 1: When v, u are perpendicular vectors, their dot product equals zero.

24 Proposition 1: When v, u are perpendicular vectors, their dot product equals zero. We can use the dot product to compute the angle between two vectors:

25 Proposition 1: When v, u are perpendicular vectors, their dot product equals zero. We can use the dot product to compute the angle between two vectors: So A B = A B cos(θ). Dot product is NEGATIVE if the angle is above 90. Dot product is POSITIVE if the angle is below 90.

26 Proposition 1: When v, u are perpendicular vectors, their dot product equals zero. We can use the dot product to compute the angle between two vectors: So A B = A B cos(θ). Dot product is NEGATIVE if the angle is above 90. Dot product is POSITIVE if the angle is below 90. These properties follow from basic trigonometry!

27 A B The quantity A B never exceeds ONE, because the cosine never does!! Thus Schwarz Inequality: A B A B

28 A B The quantity A B never exceeds ONE, because the cosine never does!! Thus Schwarz Inequality: A B A B Triangle inequality: Length of (A + B) is less than or equal to the length of A plus the length of B. A + B 2 = (A + B) (A + B) = A A + A B + A B + B B = A 2 + 2(A B) + B 2 because A B = B A A 2 +2( A B )+ B 2 = ( A + B ) 2 Schwarz Ineq. Thus A + B 2 ( A + B ) 2. Taking (positive) square root gives A + B A + B

29 The Law of Cosines can be justified with vectors, too! HOW CAN YOU SHOW THE LAW OF COSINES IS CORRECT USING VECTOR PROPERTIES?

30 Lines, Planes, and HYPER-planes, a first teaser Two points define a line (ONE dimensional object)

31 Lines, Planes, and HYPER-planes, a first teaser Two points define a line (ONE dimensional object) in R 2 given by a single equation ax + by = q, but more equations are needed in higher dimension!! Three non-collinear points define a plane (TWO dimensional object)

32 Lines, Planes, and HYPER-planes, a first teaser Two points define a line (ONE dimensional object) in R 2 given by a single equation ax + by = q, but more equations are needed in higher dimension!! Three non-collinear points define a plane (TWO dimensional object) in R 3 given by a single equation ax + by + cz = q. Again, more equations are needed to describe the same object in higher dimension!

33 Lines, Planes, and HYPER-planes, a first teaser Two points define a line (ONE dimensional object) in R 2 given by a single equation ax + by = q, but more equations are needed in higher dimension!! Three non-collinear points define a plane (TWO dimensional object) in R 3 given by a single equation ax + by + cz = q. Again, more equations are needed to describe the same object in higher dimension! Four non-collinear or co-planar points in 4-dimensions or more define a 3-hyperplane (THREE dimensional object)

34 Lines, Planes, and HYPER-planes, a first teaser Two points define a line (ONE dimensional object) in R 2 given by a single equation ax + by = q, but more equations are needed in higher dimension!! Three non-collinear points define a plane (TWO dimensional object) in R 3 given by a single equation ax + by + cz = q. Again, more equations are needed to describe the same object in higher dimension! Four non-collinear or co-planar points in 4-dimensions or more define a 3-hyperplane (THREE dimensional object) inside R 4 given by one equation ax + by + cz + dw = q!

35 Lines, Planes, and HYPER-planes, a first teaser Two points define a line (ONE dimensional object) in R 2 given by a single equation ax + by = q, but more equations are needed in higher dimension!! Three non-collinear points define a plane (TWO dimensional object) in R 3 given by a single equation ax + by + cz = q. Again, more equations are needed to describe the same object in higher dimension! Four non-collinear or co-planar points in 4-dimensions or more define a 3-hyperplane (THREE dimensional object) inside R 4 given by one equation ax + by + cz + dw = q! Challenge: Given the points how do you find the equation of the line, plane or hyperplane that contains them? How do you find the coefficients a,b,d,c,q, etc.?

36 You are given points (2, 3, 4) and (3, 2, 5). Find the parametric line equations that describe the line x = 2 + t, y = 3 5t, z = 4 + 9t. You are given points (2, 2, 1), ( 1, 0, 3), and (5, 3, 4). Find the equation of the plane that contains them?

37 You are given points (2, 3, 4) and (3, 2, 5). Find the parametric line equations that describe the line x = 2 + t, y = 3 5t, z = 4 + 9t. You are given points (2, 2, 1), ( 1, 0, 3), and (5, 3, 4). Find the equation of the plane that contains them? We are looking for ax + by + cz = d that contains those points! We need to find a, b, c, d. HOW? Learn this in 21D!! You can use the cross-product.

38 You are given points (2, 3, 4) and (3, 2, 5). Find the parametric line equations that describe the line x = 2 + t, y = 3 5t, z = 4 + 9t. You are given points (2, 2, 1), ( 1, 0, 3), and (5, 3, 4). Find the equation of the plane that contains them? We are looking for ax + by + cz = d that contains those points! We need to find a, b, c, d. HOW? Learn this in 21D!! You can use the cross-product. But suppose instead you are given (7, 2, 3, 9), (3, 4, 1, 2), (1, 0, 3, 5), (1, 1, 0, 2). How to find ax + by + cz + dw = q?

39 You are given points (2, 3, 4) and (3, 2, 5). Find the parametric line equations that describe the line x = 2 + t, y = 3 5t, z = 4 + 9t. You are given points (2, 2, 1), ( 1, 0, 3), and (5, 3, 4). Find the equation of the plane that contains them? We are looking for ax + by + cz = d that contains those points! We need to find a, b, c, d. HOW? Learn this in 21D!! You can use the cross-product. But suppose instead you are given (7, 2, 3, 9), (3, 4, 1, 2), (1, 0, 3, 5), (1, 1, 0, 2). How to find ax + by + cz + dw = q? KEY IDEA Each point must satisfy the equation ax + by + cz + dw = q, each point gives ONE linear equation! Set up a system of linear equations with variables a, b, c, d, q and then solve the system. The system is now: 7 a + 2 b 3 c + 9 d q = 0, 3 a + 4 b + c 2 d q = 0, a + 3 c 5 d q = 0, a + b + 2 d q = 0.

40 NOTE: This is the same as saying: there are numbers a, b, c, d such that a b c d = q q q q

41 NOTE: This is the same as saying: there are numbers a, b, c, d such that a b c d = q q q q If the 4 points (7, 2, 3, 9), (3, 4, 1, 2), (1, 0, 3, 5), (1, 1, 0, 2) are in a hyperplane the vector (q, q, q, q) T is a linear combination of the 4 vectors you get by looking at the first, second, third, and fourth entries of the original points!

42 To find a non-trivial solution we can add the condition that q = 1! The new system is 7 a + 2 b 3 c + 9 d q = 0, 3 a + 4 b + c 2 d q = 0, a + 3 c 5 d q = 0, a + b + 2 d q = 0, AND q = 1. We can solve any system of linear equations using MATLAB. Create a MATRIX for the system: The right-hand side vector we want to solve for is b = (0, 0, 0, 0, 1) T. Solution is (0, 1/5, 1, 2/5, 1).

43 Linear dependence and independence: A first teaser Question: If you have two vectors in the plane (a, b) and (c, d), when are they giving the same line through the origin?

44 Linear dependence and independence: A first teaser Question: If you have two vectors in the plane (a, b) and (c, d), when are they giving the same line through the origin? (a, b) must be a multiple of (c, d): (a, b) = λ(c, d) or (a, b) λ(c, d) = 0!

45 Linear dependence and independence: A first teaser Question: If you have two vectors in the plane (a, b) and (c, d), when are they giving the same line through the origin? (a, b) must be a multiple of (c, d): (a, b) = λ(c, d) or (a, b) λ(c, d) = 0! Question: Two (non-collinear) vectors (p, q, r), (u, v, w) in R 3 define a plane P through the origin (0, 0, 0). If you are given a new vector (e, f, g), how do you know whether (e, f, g) is in P?

46 Linear dependence and independence: A first teaser Question: If you have two vectors in the plane (a, b) and (c, d), when are they giving the same line through the origin? (a, b) must be a multiple of (c, d): (a, b) = λ(c, d) or (a, b) λ(c, d) = 0! Question: Two (non-collinear) vectors (p, q, r), (u, v, w) in R 3 define a plane P through the origin (0, 0, 0). If you are given a new vector (e, f, g), how do you know whether (e, f, g) is in P? Say the plane has equation ax + by + cz = 0, for some unknown numbers a, b, c (not all zero!) then it must be the case that p a u e + b q v f + c r w g = 0

47 Definition We say that vectors v 1, v 2,..., v k are linearly dependent if there are numbers λ 1,..., λ k, not all zero, such that λ 1 v 1 + λ 2 v λ k v k = 0 There is a linear combination that gives the zero vector!

48 Definition We say that vectors v 1, v 2,..., v k are linearly dependent if there are numbers λ 1,..., λ k, not all zero, such that λ 1 v 1 + λ 2 v λ k v k = 0 There is a linear combination that gives the zero vector! If such numbers do not exist then we say the vectors are linearly independent E.g., (1, 0, 0), (0, 1, 0), (0, 0, 1).

49 Definition We say that vectors v 1, v 2,..., v k are linearly dependent if there are numbers λ 1,..., λ k, not all zero, such that λ 1 v 1 + λ 2 v λ k v k = 0 There is a linear combination that gives the zero vector! If such numbers do not exist then we say the vectors are linearly independent E.g., (1, 0, 0), (0, 1, 0), (0, 0, 1). Collinear vectors (all on one line) and co-planar vectors (all on one plane) are special cases of linearly dependent vectors!

50 Definition We say that vectors v 1, v 2,..., v k are linearly dependent if there are numbers λ 1,..., λ k, not all zero, such that λ 1 v 1 + λ 2 v λ k v k = 0 There is a linear combination that gives the zero vector! If such numbers do not exist then we say the vectors are linearly independent E.g., (1, 0, 0), (0, 1, 0), (0, 0, 1). Collinear vectors (all on one line) and co-planar vectors (all on one plane) are special cases of linearly dependent vectors! HOW to find such numbers λ i? Again we are solving a system of linear equations!

51 Matrices (1.3) A MATRIX is an m n array of numbers. It has m rows and n columns

52 Matrices (1.3) A MATRIX is an m n array of numbers. It has m rows and n columns NOTE: Vectors are matrices (n 1 matrices).

53 Matrices (1.3) A MATRIX is an m n array of numbers. It has m rows and n columns NOTE: Vectors are matrices (n 1 matrices). Matrices can be thought of as groups of vectors!

54 Matrices (1.3) A MATRIX is an m n array of numbers. It has m rows and n columns NOTE: Vectors are matrices (n 1 matrices). Matrices can be thought of as groups of vectors! E.g., the above matrix is made of FIVE vectors inside 4-dimensional space.

55 Matrix multiplication How can we write a linear combination of vectors in terms of matrices? Multiplication of a matrix and a vector!

56 Matrix multiplication How can we write a linear combination of vectors in terms of matrices? Multiplication of a matrix and a vector! We think of λ 1 p 1 + λ 2 p λ n p n as a matrix with n columns A = [p 1 p 2... p n ] λ 2 multiplied by the vector u = of n entries (rows!). λ n This is just written as Au = λ 1 p 1 + λ 2 p λ n p n λ 1

57 Matrix multiplication How can we write a linear combination of vectors in terms of matrices? Multiplication of a matrix and a vector! We think of λ 1 p 1 + λ 2 p λ n p n as a matrix with n columns A = [p 1 p 2... p n ] λ 2 multiplied by the vector u = of n entries (rows!). λ n This is just written as Au = λ 1 p 1 + λ 2 p λ n p n Example: ( ) 9 8 = 9 7 λ 1 ( ) ( ) ( ) ( ) 3 46 = 6 118

58 Matrix multiplication How can we write a linear combination of vectors in terms of matrices? Multiplication of a matrix and a vector! We think of λ 1 p 1 + λ 2 p λ n p n as a matrix with n columns A = [p 1 p 2... p n ] λ 2 multiplied by the vector u = of n entries (rows!). λ n This is just written as Au = λ 1 p 1 + λ 2 p λ n p n Example: ( ) 9 8 = 9 7 λ 1 ( ) ( ) ( ) ( ) 3 46 = 6 118

59 General Matrix Multiplication Now we extend the same principle to multiply two matrices A, B. Key Point Multiplication of two matrices can be done vector by vector! That is IF the sizes of the matrices match!

60 General Matrix Multiplication Now we extend the same principle to multiply two matrices A, B. Key Point Multiplication of two matrices can be done vector by vector! That is IF the sizes of the matrices match! For an m n matrix A and an n p matrix B the multiplication works as follows:

61 General Matrix Multiplication Now we extend the same principle to multiply two matrices A, B. Key Point Multiplication of two matrices can be done vector by vector! That is IF the sizes of the matrices match! For an m n matrix A and an n p matrix B the multiplication works as follows: Say B has column vectors B = [v 1 v 2 v 3... v p ] The new matrix AB is a matrix with columns (matching the right size for A). AB = [Av 1 Av 2 Av 3... Av p ]

62 General Matrix Multiplication Now we extend the same principle to multiply two matrices A, B. Key Point Multiplication of two matrices can be done vector by vector! That is IF the sizes of the matrices match! For an m n matrix A and an n p matrix B the multiplication works as follows: Say B has column vectors B = [v 1 v 2 v 3... v p ] The new matrix AB is a matrix with columns (matching the right size for A). AB = [Av 1 Av 2 Av 3... Av p ]

63 General Matrix Multiplication Say B has column vectors B = [v 1 v 2 v 3... v p ] The new matrix AB is a matrix with columns (matching the right size for A). AB = [Av 1 Av 2 Av 3... Av p ] Example: A = B =

64 General Matrix Multiplication Say B has column vectors B = [v 1 v 2 v 3... v p ] The new matrix AB is a matrix with columns (matching the right size for A). AB = [Av 1 Av 2 Av 3... Av p ] Example: A = B = Careful! We cannot multiply any pair of matrices and AB is not always equal to BA!

65 1 2 3 AB = There are several useful operations and properties of matrix multiplication: Multiplication of a matrix A by a number r: r times each individual entry in A. (AB)C=A(BC) A(B+C)=AB+AC (A+B)C=AC+BC Operations are OK, as long as sizes of matrices match!