Vector and Matrix Introduction
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1 Vector and Matrix Introduction June 13, Vectors A scalar is a magnitude, or just a number An element of R is a scalar Until this point, most variables you have used denoted scalars Quantities measured in scalars include speed, length, and mass A vector is a magnitude with a direction We write vectors in R n as a = a 1, a 2,, a n, where a 1, a 2,, a n R are scalars We call a 1, a 2,, a n the components of a Quantities measured by vectors include velocity, displacement, and force Example: Suppose you are traveling west at 10 mph We say your (scalar) speed is 10 mph If we orient the Cartesian plane so that the positive x-axis points east and let units be miles per hour, then your (vector) velocity is 10, 0 To show that a symbol stands for a vector, we underline it (a) or make it bold (a) Some authors write an arrow over the top of it ( a ) For example, if you see the expression au + 2bv, you may assume that a and b are scalars, while u and v are vectors 11 Geometric Interpretation Geometrically, we may view a vector a 1, a 2 as an arrow in the Cartesian plane from the point (0, 0) to the point (a 1, a 2 ) For example, the vector 1, 2 will point diagonally left and down, and the vector 3, 1 will point left and up (3,1) (0,0) (1,-2) 12 Scalar Multiplication In R 2, the vectors a = 1, 1 and b = 2, 2 point in the same direction but have a different length We say these vectors are scalar multiples because if we multiply each component of a by the same scalar, 2, we obtain b 1
2 In general, to multiply a vector a = a 1, a 2,, a n by a scalar c, we multiply each component by the scalar 13 Length of a Vector ca = c a 1, a 2,, a n = ca 1, ca 2,, ca n To find the magnitude of the vector a 1, a 2 R 2, we can use the geometric interpretation The Pythagorean Theorem tells us that the length of the line segment from (0, 0) to (a 1, a 2 ) is a a 2 2 In general, we denote the length or magnitude of the vector a = a 1, a 2,, a n by a, and we compute it as follows: a = a 1, a 2,, a n = a a a 2 n 14 Unit Vectors To do certain computations later in this course, we will need a vector of length one in a given direction We call this a unit vector To find a unit vector in the direction of a, we divide a by its magnitude: a a = 1 a a Example: The unit vector in the direction of the vector 2, 4, 1 is 2, 4, 1 2, 4, 1 = = 2 4,, 1 = , 4, , , Addition and Subtraction of Vectors Given two vectors a = a 1, a 2,, a n and b = b 1, b 2,, b n, we may add and subtract them componentwise a 1, a 2,, a n + b 1, b 2,, b n = a 1 + b 1, a 2 + b 2,, a n + b n Example: a 1, a 2,, a n b 1, b 2,, b n = a 1 b 1, a 2 b 2,, a n b n 1, 4, 5, 3 + 2, 3, 6, 5 2, 4, 3, 2 = 3, 5, 14, 10 We may find the vector between two points by taking the difference in the coordinates of the points The vector from the point P (p 1, p 2,, p n ) to the point Q(q 1, q 2,, q n ) is P Q = q 1 p 1, q 2 p 2,, q n p n For example, the vector in the direction from the point P (2, 4, 1) to the point Q( 5, 4, 6) is P Q = 7, 0, 7 2
3 16 Standard basis vectors Alternatively, we can write a vector in R 2 using standard basis vectors, which are defined by i = 1, 0 and j = 0, 1 Using standard basis vectors, we can write a 1, a 2 as a 1 i + a 2 j Similarly, in R 3 we define standard basis vectors i = 1, 0, 0, j = 0, 1, 0, and k = 0, 0, 1 Examples: 3, 0, 2 = 3i + 2k 4, 1 = 4i j 17 Properties We have the following properties for vector addition and scalar multiplication We write 0 for the vector whose components are all 0 1 a + b = b + a 2 a + (b + c) = (a + b) + c 3 a + 0 = a 4 a + ( 1)a = 0 5 c(a + b) = ca + cb 6 (c + d)a = ca + da 7 (cd)a = c(da) 8 1a = a 18 Dot Product Given two vectors a, b R n, we define their dot product as follows: a b = a 1, a 2,, a n b 1, b 2,, b n = a 1 b 1 + a 2 b a n b n Notice that the dot product of two vectors is a scalar Examples: 4, 3 5, 6 = (4)( 5) + ( 3)(6) = 38 (2i + j 3k) (4i j 2k) = (2)(4) + (1)( 1) + ( 3)( 2) = 13 We have the following properties of dot products 1 a a = a 2 2 a b = b a 3 a (b + c) = a b + a c 4 (ca) b = c(a b) = a (cb) 5 0 a = 0 3
4 19 Theorems The following theorem explains how the dot product gives us information about the angle between two vectors Theorem: If θ is the angle between vectors a and b with 0 θ π, then a b = a b cos θ Example: Find the angle between the vectors 4, 1, 3 and 2, 0, 1 a b = a b cos θ 4, 1, 3 2, 0, 1 = 4, 1, 3 2, 0, 1 cos θ 5 = 26 5 cos θ = cos θ ( ) 5 θ = arccos 130 ( ) 130 θ = arccos 26 We say two vectors are orthogonal if the angle between them is π/2 This is the same as being perpendicular Theorem: Two vectors a and b are orthogonal if and only if a b = 0 Example: Determine whether each pair of vectors is parallel, perdicular, or neither (a) a = 4i + 2j and b = 2i j a b = a b cos θ (4i + 2j) ( 2i j) = 4i + 2j 2i j cos θ 10 = ( 20)( 5) cos θ 10 = 10 cos θ 1 = cos θ θ = π The angle between a and b is π, so the vectors are parallel (b) a = 6, 0, 9 and b = 3, 8, 2 a b = 6, 0, 9 3, 8, 2 = 0 The dot product of a and b is 0, so by the theorem they are orthogonal (or perpendicular) 4
5 2 Matrices We will introduce matrices and discuss the basic properties and operations that will be needed for this course Matrices are studies in greater depth in linear algebra courses A matrix is a way of organizing information into columns and rows to form a rectangle We could also define it as an extension of a vector, where values appear both across and down Here are some examples of matrices (in different notations) M = N = [ ] L = Notice that we may use round or square brackets; they mean the same thing We say that M is a 3 3 matrix, N is a 2 3 matrix, and L is a 3 1 matrix In general, an m n matrix has m horizontal rows and n vertical columns A matrix (like M) with the same number of columns and rows is called a square matrix We write a general matrix with double subscripts Here is a general m n matrix a 1,1 a 1,2 a 1,n a 2,1 a 2,2 a 2,n A = a m,1 a m,2 a m,n We may also write this general matrix as A = {a i,j } 1 i m,1 j n To view a vector a 1, a 2,, a n as a matrix, we write: a 1 a 2 a 1, a 2,, a n = a n 21 Tanspose The transpose of a matrix is the same matrix with its columns and rows exchanged We denote the transpose of a matrix M by M T If A is the general matrix given above, then its transpose is A T = a 1,1 a 2,1 a m,1 a 1,2 a 2,2 a m,2 a 1,n a 2,n a m,n
6 Example: Let M be the 4 2 matrix 2 3 M = Then M T is the 2 4 matrix 22 Determinant M T = ( If M is a square matrix, we may compute an associated real number called its determinant, which is denoted M First, suppose M is a 2 2 matrix Then its determinant computed as follows: M = a b c d = ad bc ( ) 1 4 Example: The determinant of M = is M = (1)( 3) (4)(2) = Second, suppose M is a 3 3 matrix Then its determinant is computed as follows: a b c M = d e f g h i = a e f h i + b f d i g + c d e g h ) = a(ei fh) + b(fg di) + c(dh eg) The last line of this looks ugly, so I suggest memorizing the 3 3 determinant in terms of the 2 2 determinants, as it is written in the first line Example: Let M = Then the determinant of M is M = = 1(8 2) 4( 5 12) + 0( 6 10) = 1(6) 4( 17) + 0 = = 74 Finally, let M be any square n n matrix Then the determinant of M is m 1,1 m 1,2 m 1,n m 2,1 m 2,2 m 2,n M = m n,1 m n,2 m n,n 6
7 = m 1,1 m 2,2 m 2,n m n,2 m n,n m 2,3 m 2,n m 2,1 m n,3 m n,n m n,1 m 2,1 m 2,n m 1,n m n,1 m n,n 1 +m 1,2 +m 1,3 m 2,4 m 2,n m 2,1 m 2,2 m n,3 m n,n m n,1 m n,2 23 Multiplication of matrices The last matrix operation we will need for this course is matrix multiplication First we will multiply a matrix by a vector Let A be an m n matrix, and let B be the matrix of a vector in R n (so B is a n 1 matrix) Then the product AB is computed as follows: AB = a 1,1 a 1,2 a 1,n a 2,1 a 2,2 a 2,n a m,1 a m,2 a m,n b 1 b 2 b n = a 1,1 b 1 + a 1,2 b a 1,n b n a 2,1 b 1 + a 2,2 b a 2,n b n a m,1 b 1 + a m,2 b a m,n b n We may think of taking the dot product of B with each row of A Notice that multiplying an m n matrix with an n 1 matrix gave us an m 1 matrix Example: = (1)(3) + ( 5)( 2) + (4)( 1) (1)(3) + (2)( 2) + ( 4)( 1) (0)(3) + ( 1)( 2) + (5)( 1) = Now we will multiply general matrices Let A be an m n matrix, and let B be an n k matrix To find the product AB, view each column of B as a vector a 1,1 a 1,2 a 1,n b 1,1 b 1,2 b 1,k a 2,1 a 2,2 a 2,n b 2,1 b 2,2 b 2,k AB = a m,1 a m,2 a m,n b n,1 b n,2 b n,k = a 1,1 a 1,2 a 1,n a 2,1 a 2,2 a 2,n a m,1 a m,2 a m,n b 1,1 b 2,1 b n,1 b 1,2 b 2,2 b n,2 b 1,k b 2,k b n,k The first column of the product AB is the product of the first column of B with A, the second column of AB is the product of the second column of B with A, etc Here is the general formula for AB 7
8 AB = Example: a 1,1 b 1,1 + a 1,2 b 2,1 + + a 1,n b n,1 a 1,1 b 1,k + a 1,2 b 2,k + + a 1,n b n,k a 2,1 b 1,1 + a 2,2 b 2,1 + + a 2,n b n,1 a 2,1 b 1,k + a 2,2 b 2,k + + a 2,n b n,k a m,1 b 1,1 + a m,2 b 2,1 + + a m,n b n,1 a m,1 b 1,k + a m,2 b 2,k + + a m,n b n,k (3)(0) + ( 2)( 1) + (1)( 5) + (4)( 3) (3)( 1) + ( 2)(2) + (1)(0) + (4)(4) = ( 1)(0) + (0)( 1) + (5)( 5) + ( 3)( 3) ( 1)( 1) + (0)(2) + (5)(0) + ( 3)(4) (2)(0) + (7)( 1) + ( 1)( 5) + (0)( 3) (2)( 1) + (7)(2) + ( 1)(0) + (0)(4) 15 9 = Observe that we could not have multiplied these two matrices in the reverse order! That is, if we attempted to find , the columns of the second matrix could not be multiplied by the rows of the first In general, the matrix product AB is defined ONLY when the number of columns of A equals the number of rows of B Notice that the product AB has the same number of rows as A and the same number of columns as B Here is the rule for dimensions in matrix multiplication: (m n matrix )(n k matrix ) = (m k matrix ) 8
Figure 1.1 Vector A and Vector F
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