Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 3 Linear Algebra Lecture 7 Matrix
1 Introduction - Algebra & Geometry - Vector - Change of coordinates (transformation)
H. W. (Due Apr. 16th) Chapter 3 3-2. 8, 9 3-3. 1 3-5. 12, 16, 21, 37 3-6. 6, 15 3-7. 23, 24 “The problems similar to the above ones will appear in the midterm exam.”
2. Matrix; row reduction (행렬 ; 행줄이기) - A matrix is just a rectangular array of quantities, usually enclosed in large parentheses. - Transpose of a matrix
- Augmented matrix (a) Eliminate the x terms in the other two equations by using the first equation. ex. 2) – 1) x 3 2) , 3) – 1) 3) (b) For convenience, interchange the second and third equations
(c) Eliminate the y terms by using the second equation. ex. 3) – 2) x 5 3) (d) Eliminate the z terms by using the third equation. ex. 1) + 3) / 11 1) , 2) – 3) / 11 2)
(e) finalizing - Allowed rules i. Interchange two rows ii. Multiply (or divide) a row by a (nonzero) constant iii. Add a multiple of one row to another; this includes subtracting, that is, using a negative multiple.
Ex. 2 We can not get an answer. The equations are inconsistent.
- Rank of a Matrix - The number of nonzero rows remaining when a matrix has been row reduced is called the rank of a matrix. - For example 2, the rank of A is 3, but the rank of M is 2. In this case, the equations are inconsistent [(rank of M) < (rank of A)]. ‘Inconsistent’ rank: 2 rank: 3
a. If (rank M) < (rank A), the equations are inconsistent and there is no solution. b. If (rank M) = (rank A) = n (number of unknowns), there is one solution. c. If (rank M) = (rank A) < n , then R unknown can be found in terms of the remaining n – R unknowns.
3. Determinants; Cramer’s rule (행렬식 ; Cramer의 규칙) - We have said that a matrix is simply a display of a set of numbers; it does not have numerical value. For a square matrix, however, there is a useful number called the determinant of the matrix. - Evaluating determinants i) 2 by 2
ii) nth order matrix - When removing the row and the column containing the element a_ij, we have the remaining determinant, M_ij, called a minor of a_ij. ex) (n-1) by (n-1) determinant
- Sign - cofactor : - Finally, multiply each element of one row (or one column) by its cofactor and add the results. “Laplace’s development”
ex) i) method 1 (third column) ii) method 2 (first row)
- Useful facts about determinants 1. If each element of one row (or one column) of a determinant is multiplied by a number k, the value of the determinant is multiplied by k. 2. The value of a determinant is zero if, (a) all elements of one row are zero (b) two rows ( or two columns) are identical (c) two rows (or two columns) are proportional. 3. If two rows ( or two columns) of a determinant are interchanged, the value of the determinant changes sign. 4. The value of a determinant is unchanged if (a) rows are written as columns are columns as rows (b) we add to each element of one row, k times the corresponding element of another row, where k is any number (and a similar statement for columns).
Example 2. Find the equation of a plane through the three given points (0,0,0), (1,2,5), and (2,-1,0)
Example 4. Evaluate the determinant 1. Subtract 4 times the fourth column from the first column, and subtract 2 times the fourth column from the third column 2. Do a Laplace development using the third row 3. Add the second row to the third row. 4. Do a Laplace development using the third row.
- Cramer’s rule - Denominator: determinant of the matrix with coefficients in the left side (M) - Numerator: for x, replace x part in M with right side part, and take the determinant. for y, replace y part in M with right side part, and take the determinant. - We can use this method when you get the solution for the n linear equations (in case that D 0).
- Rank of a Matrix To find the rank of a matrix, we look at all the square submatrices and find their determinants. The order of the largest nonzero determinant is the rank of the matrix. cf. submatrix: a matrix remaining if we remove some rows and/or columns. Example 6. The submatrices of the original matrix are four (123, 124, 134, and 234). Because the first and the second columns are absolutely the same, the determinants of 123 and 123 should be zero. The absolute values of 134 and 234 should be the same. For this reason, we need to check only 134. “The rank should be less than 3.”
Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 3 Linear Algebra Lecture 8 Vector
4. Vector (벡터) - Notation - Magnitude - Addition of vectors
- Multiplication by a constant & subtraction - Unit vector
- Angles between two vectors using scalar product example) Find the angles between these two vectors
5. Lines and planes (직선과 평면) - A great deal of analytic geometry can be simplified by the use of vector notation. Such things as equations of lines and planes, and distances between points or between lines and planes often occur in physics, and it is very useful to be able to find them quickly. Vector notation will help you do these more easily. ‘points vectors’
- Straight Lines To determine a specific line, we need one point and a slope (= two points).
- Planes To determine a specific plane, we need a point and a normal vector.
Example 1. Find the equation of the plane through the three points A(-1,1,1), B(2,3,0), C(0,1,-2). - Because points are given, what we have to do is to find the normal vector. C N AC B AB A
Example 1. Find the equation of the plane through the three points A(-1,1,1), B(2,3,0), C(0,1,-2).
Example 2 Find the equation of a line through (1,0,-2) and perpendicular to the plane of Example 1
Example 3 Find the distance from the point P(1,-2,3) to the plane 3x-2y+z+1=0. (Use the dot product.) P PR: distance we want to know Q : any point on the plane Q R - Choose Q in the easiest way, e.g., Q=(0,0.-1) or (1,2,0) (as in the text)
Example 4 Find the distance from P(1,2,-1) to the line joining P1(0,0,0) and P2(-1,0,2). (Use the cross product.)
Example 5 Find the distance between the lines, r=i-2j+(i-k)t, r=2j-k+(j-i)t. P A n Q B
Example 6. Find the direction of the lines of intersection of the planes Note) the intersection lines are perpendicular to both the planes. Example 7. Find the cosine of the angle between two planes Note) The angle between the planes is the same as the angle between the normal vectors to the planes
Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 3 Linear Algebra Lecture 9 Matrix operation
6. Matrix operations (행렬계산) - Matrix equations - Multiplication of a matrix by a number
- Addition of Matrices Note) Addition (or subtraction) can be done with the same type of matrices
- Multiplication of matrices The element in row i and column j of the product matrix AB is equal to row i of A times column j of B. [(# of row of A) = (# of column j)] In index notation, ex. Example 1
Example 2. Find AB and BA “not commutative”
- Zero matrix : zero or null matrix means one with all its elements equal to zero. cf. - Identity matrix or Unit matrix
- Operation with Determinants - Applications of matrix multiplication
Example 3. Finding the cofactor