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# Midterm I Review

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### Midterm I Review

1. 1. Review for Midterm I Math 20 October 16, 2007 Announcements Midterm I 10/18, Hall A 7–8:30pm ML Oﬃce Hours Wednesday 1–3 (SC 323) SS review session Wednesday 8–9pm (location TBA) Old exams and solutions on website
2. 2. Outline Determinants by sudoku Vector and Matrix Algebra patterns Vectors Determinants by Cofactors Algebra of vectors Systems of Linear Equations Scalar Product Matrices Inversion Addition and scalar Determinants and the multiplication of matrices inverse Inverses of 2 × 2 matrices Matrix Multiplication The Transpose Computing inverses with the Geometry adjoint matrix Lines Computing inverses with row Planes reduction Determinants Properties of the inverse
3. 3. Vector and Matrix Algebra Learning Objectives Add, scale, and compute linear combinations of vectors Compute the scalar product of two vectors Add and scale matrices Multiply matrices all of these with the caveat of “if possible.”
4. 4. Vectors Deﬁnition An n-vector (or simply vector) is an ordered list of n numbers. We can write them in a row or in a column.  1 b = (1, 0, −1) 2 a= 3 In linear algebra we mostly work with column vectors.
5. 5. Algebra of vectors Deﬁnition Addition of vectors is deﬁned componentwise, as is scalar multiplication:     a1 b1 a1 + b 1  a2   b 2   a2 + b 2   . + . = .      . .  .  . . . an bn an + bn    a1 ta1 a2  ta2  t . = .     .  .  . . an tan
6. 6. Dot product Deﬁnition Given two vectors of the same dimension (size), their scalar product (or “dot product”) is the sum of the product of the components of the vectors:    a1 b1 n a2  b2   .  ·  .  = a1 b1 + a2 b2 + · · · + an bn = ai bi    . . . . i=1 an bn
7. 7. Matrices Deﬁnition An m × n matrix is a rectangular array of mn numbers arranged in m horizontal rows and n vertical columns.   a11 a12 · · · a1j · · · a1n  a21 a22 · · · a2j · · · a2n    . . . . .. .. . . . . . . . . . . A=  ai1 ai2 · · · aij · · · ain    . . . . .. .. . . . . . . . . . . am1 am2 · · · amj · · · amn
8. 8. Addition and scalar multiplication of matrices Matrices can be added and scaled just like vectors: Example 1 −1 12 21 + = 34 02 36 Example 11 44 4 = −1 2 −4 8
9. 9. The matrix-vector product Deﬁnition   v1 v  Let A = (aij ) be an m × n matrix and v =  2  a n-vector . . .  vn (column vector). The matrix-vector product of A and v is the  w1  w2  vector Av =  , where . . . wm n wk = ak1 v1 + ak2 v2 + · · · + akn vn = akj vj , j=1 the dot product of kth row of A with v.
10. 10. Example Let   23 2 A = −1 4 and v= −1 03 Find Av.
11. 11. Example Let   23 2 A = −1 4 and v= −1 03 Find Av. Solution   2 · 2 + 3 · (−1) Av = (−1) · 2 + 4 · (−1) 0 · 2 + 3 · (−1)
12. 12. Example Let   23 2 A = −1 4 and v= −1 03 Find Av. Solution    2 · 2 + 3 · (−1) 4−1 Av = (−1) · 2 + 4 · (−1) = −2 − 4 0 · 2 + 3 · (−1) 0−3
13. 13. Example Let   23 2 A = −1 4 and v= −1 03 Find Av. Solution    2 · 2 + 3 · (−1) 4−1 1 (−1) · 2 + 4 · (−1) = −2 − 4 = −6 . Av = 0 · 2 + 3 · (−1) 0−3 −3
14. 14. Matrix product, deﬁned Deﬁnition Let A be an m × n matrix and B a n × p matrix. Then the matrix product of A and B is the m × p matrix whose jth column is Abj . In other words, the (i, j)th entry of AB is the dot product of ith row of A and the jth column of B. In symbols n (AB)ij = aik bkj . k=1 Example     1.5 0.5 1  125 115 110 105   0 0.25 0 ‘70 60 50 40 5 7.5 7.5 7.5      1.5 0.25 0  20 30 30 30 = 100 97.5 82.5 67.5     2 2 3 10 10 20 30 210 210 220 230  3 2 2 270 260 250 240
15. 15. The Transpose There is another operation on matrices, which is just ﬂipping rows and columns. Deﬁnition Let A = (aij )m×n be a matrix. The transpose of A is the matrix A = (aij )n×m whose (i, j)th entry is aji .
16. 16. The Transpose There is another operation on matrices, which is just ﬂipping rows and columns. Deﬁnition Let A = (aij )m×n be a matrix. The transpose of A is the matrix A = (aij )n×m whose (i, j)th entry is aji . Example  12 Let A = 3 4. Then 56 A=
17. 17. The Transpose There is another operation on matrices, which is just ﬂipping rows and columns. Deﬁnition Let A = (aij )m×n be a matrix. The transpose of A is the matrix A = (aij )n×m whose (i, j)th entry is aji . Example  12 Let A = 3 4. Then 56 135 A= . 246
18. 18. The Transpose There is another operation on matrices, which is just ﬂipping rows and columns. Deﬁnition Let A = (aij )m×n be a matrix. The transpose of A is the matrix A = (aij )n×m whose (i, j)th entry is aji . Example  12 Let A = 3 4. Then 56 135 A= . 246 Fact Given matrices A and B, of suitable dimensions, (AB) = B A
19. 19. Outline Determinants by sudoku Vector and Matrix Algebra patterns Vectors Determinants by Cofactors Algebra of vectors Systems of Linear Equations Scalar Product Matrices Inversion Addition and scalar Determinants and the multiplication of matrices inverse Inverses of 2 × 2 matrices Matrix Multiplication The Transpose Computing inverses with the Geometry adjoint matrix Lines Computing inverses with row Planes reduction Determinants Properties of the inverse
20. 20. Geometry Learning Objectives Given a point and a line, decide if the point is on the line Given a point and a direction, or two points, ﬁnd the equation for the line Given a point and a plane, decide of the point is on the plane Given a point and a normal vector, ﬁnd the equation for the plane
21. 21. Lines Deﬁnition The line L through (the head of) a parallel to v is the set of all x such that x = a + tv for some real number t. The line L through (the heads of) a and b is x = a + t(b − a) = (1 − t)a + tb
22. 22. Example Find an equation for the line through (−2, 0, 4) parallel to (2, 4, −2)
23. 23. Example Find an equation for the line through (−2, 0, 4) parallel to (2, 4, −2) Solution x = (−2, 0, 4) + t(2, 4, −2)
24. 24. Example Find an equation for the line through (−3, 2, −3) and (1, −1, 4)
25. 25. Example Find an equation for the line through (−3, 2, −3) and (1, −1, 4) Is (1, 2, 3) on this line?
26. 26. Example Find an equation for the line through (−3, 2, −3) and (1, −1, 4) Is (1, 2, 3) on this line? Answer. No!
27. 27. Planes Deﬁnition A hyperplane through (the head of) a that is orthogonal to a vector p is the set of all (heads of) vectors x such that p · (x − a) = 0
28. 28. Example Find an equation for the plane through (−3, 0, 7) perpendicular to (5, 2, −1)
29. 29. Example Find an equation for the plane through (−3, 0, 7) perpendicular to (5, 2, −1) Solution 5x + 2y − z = 22
30. 30. Example Find an equation for the plane through (−3, 0, 7) perpendicular to (5, 2, −1) Solution 5x + 2y − z = 22 Question Is (1, 2, 3) on this plane? Is there a point on the plane with z = 0?
31. 31. Fact If the equation for a plane is Ax + By + Cz = D, then p = (A, B, C ) is normal to the plane.
32. 32. Outline Determinants by sudoku Vector and Matrix Algebra patterns Vectors Determinants by Cofactors Algebra of vectors Systems of Linear Equations Scalar Product Matrices Inversion Addition and scalar Determinants and the multiplication of matrices inverse Inverses of 2 × 2 matrices Matrix Multiplication The Transpose Computing inverses with the Geometry adjoint matrix Lines Computing inverses with row Planes reduction Determinants Properties of the inverse
33. 33. Determinants Learning Objectives Compute determinants of n × n matrices.
34. 34. The determinant Deﬁnition a11 a12 The determinant of a 2 × 2 matrix A = is the number a21 a22 a11 a12 = a11 a22 − a21 a12 a21 a22
35. 35. Deﬁnition The determinant of a 3 × 3 matrix is a11 a12 a13 a21 a22 a23 = a11 a22 a33 − a11 a23 a32 − a21 a12 a33 a31 a32 a33 + a21 a13 a32 + a31 a12 a23 − a31 a22 a13
36. 36. Sarrus’s Rule
37. 37. Sarrus’s Rule a11 a12 a13 a11 a22 a33 + a12 a23 a31 = +a13 a21 a32 − a13 a22 a31 a21 a22 a23 −a11 a23 a32 − a12 a21 a33 a31 a32 a33
38. 38. Sarrus’s Rule a11 a12 a13 a11 a12 a11 a22 a33 + a12 a23 a31 = +a13 a21 a32 − a13 a22 a31 a21 a22 a23 a21 a22 −a11 a23 a32 − a12 a21 a33 a31 a32 a33 a31 a32
39. 39. Sarrus’s Rule a11 a12 a13 a11 a12 a11 a22 a33 + a12 a23 a31 = +a13 a21 a32 − a13 a22 a31 a21 a22 a23 a21 a22 −a11 a23 a32 − a12 a21 a33 a31 a32 a33 a31 a32
40. 40. Sarrus’s Rule a11 a12 a13 a11 a12 a11 a22 a33 + a12 a23 a31 = +a13 a21 a32 − a13 a22 a31 a21 a22 a23 a21 a22 −a11 a23 a32 − a12 a21 a33 a31 a32 a33 a31 a32
41. 41. Sarrus’s Rule a11 a12 a13 a11 a12 a11 a22 a33 + a12 a23 a31 = +a13 a21 a32 − a13 a22 a31 a21 a22 a23 a21 a22 −a11 a23 a32 − a12 a21 a33 a31 a32 a33 a31 a32 This trick does not work for any other determinants!
42. 42. Determinants by sudoku patterns Deﬁnition Let A = (aij )n×n be a matrix. The determinant of A is a sum of all products of n elements of the matrix, where each product takes exactly one entry from each row and column.
43. 43. Determinants by sudoku patterns Deﬁnition Let A = (aij )n×n be a matrix. The determinant of A is a sum of all products of n elements of the matrix, where each product takes exactly one entry from each row and column. The sign of each product is given by (−1)σ , where σ is the number of upwards lines used when all the entries in a pattern are connected.
44. 44. 4 × 4 sudoku patterns − − − + + + − − − + + + − − − + + + − − − + + +
45. 45. Determinants by Cofactors Deﬁnition Let A = (aij )n×n be a matrix. The (i, j)-minor of A is the matrix obtained from A by deleting the ith row and j column. This matrix has dimensions (n − 1) × (n − 1). The (i, j) cofactor of A is the determinant of the (i, j) minor times (−1)i+j .
46. 46. The signs here seem complicated, but they’re not. The number (−1)i+j is 1 if i and j are both even or both odd, and −1 otherwise. They make a checkerboard pattern:   + − + ··· − + − · · ·   + − + · · ·   . . . .. ... . ...
47. 47. Fact The determinant of A = (aij )n×n is the sum a11 C11 + a12 C12 + · · · + a1n C1n
48. 48. Fact The determinant of A = (aij )n×n is the sum a11 C11 + a12 C12 + · · · + a1n C1n Fact The determinant of A = (aij )n×n is the sum a11 Ci1 + ai2 Ci2 + · · · + ain Cin for any i.
49. 49. Fact The determinant of A = (aij )n×n is the sum a11 C11 + a12 C12 + · · · + a1n C1n Fact The determinant of A = (aij )n×n is the sum a11 Ci1 + ai2 Ci2 + · · · + ain Cin for any i. Fact The determinant of A = (aij )n×n is the sum a1j C1j + a2j C2j + · · · + anj Cnj for any j.
50. 50. Theorem (Rules for Determinants) Let A be an n × n matrix. 1. If a row or column of A is full of zeros, then |A| = 0. 2. |A | = |A| 3. If B is the matrix obtained by multiplying each entry of one row or column of A by the same number α, then |B| = α |A|. 4. If two rows or columns of A are interchanged, then the determinant changes its sign but keeps its absolute value. 5. If a row or column of A is duplicated, then |A| = 0.
51. 51. Theorem (Rules for Determinants, continued) Let A be an n × n matrix. 5. If a row or column of A is proportional to another, then |A| = 0. 6. If a scalar multiple of one row (or column) of A is added to another row (or column), then the determinant does not change. 7. The determinant of the product of two matrices is the product of the determinants of those matrices: |AB| = |A| |B| 8. if α is any real number, then |αA| = αn |A|.
52. 52. Outline Determinants by sudoku Vector and Matrix Algebra patterns Vectors Determinants by Cofactors Algebra of vectors Systems of Linear Equations Scalar Product Matrices Inversion Addition and scalar Determinants and the multiplication of matrices inverse Inverses of 2 × 2 matrices Matrix Multiplication The Transpose Computing inverses with the Geometry adjoint matrix Lines Computing inverses with row Planes reduction Determinants Properties of the inverse
53. 53. Systems of Linear Equations Learning Objectives Solve a System of Linear Equations using Gaussian Elimination Find the (R)REF of a matrix.
54. 54. Row Operations The operations on systems of linear equations are reﬂected in the augmented matrix. 1. Transposing (switching) rows in an augmented matrix does not change the solution. 2. Scaling any row in an augmented matrix does not change the solution. 3. Adding to any row in an augmented matrix any multiple of any other row in the matrix does not change the solution.
55. 55. Gaussian Elimination 1. Locate the ﬁrst nonzero column. This is pivot column, and the top row in this column is called a pivot position. Transpose rows to make sure this position has a nonzero entry. If you like, scale the row to make this position equal to one. 2. Use row operations to make all entries below the pivot position zero. 3. Repeat Steps 1 and 2 on the submatrix below the ﬁrst row and to the right of the ﬁrst column. Finally, you will arrive at a matrix in row echelon form. (up to here is called the forward pass) 4. Scale the bottom row to make the leading entry one. 5. Use row operations to make all entries above this entry zero. 6. Repeat Steps 4 and 5 on the submatrix formed above and to the left of this entry. (These steps are called the backward pass)
56. 56. Example Solve the system of linear equations 2x+ 8y +4z =2 2x+ 5y + z =5 4x+10y −4z =1
57. 57. Outline Determinants by sudoku Vector and Matrix Algebra patterns Vectors Determinants by Cofactors Algebra of vectors Systems of Linear Equations Scalar Product Matrices Inversion Addition and scalar Determinants and the multiplication of matrices inverse Inverses of 2 × 2 matrices Matrix Multiplication The Transpose Computing inverses with the Geometry adjoint matrix Lines Computing inverses with row Planes reduction Determinants Properties of the inverse
58. 58. Inversion Learning Objectives Use the determinant to determine whether a matrix is invertible Find the inverse of a matrix
59. 59. Determinants and the inverse If A has an inverse A−1 , what is |A|−1 ? Answer. |A| A−1 = AA−1 = |I| = 1 Fact A is invertible if and only if |A| = 0.
60. 60. Inverses of 2 × 2 matrices ab Let A = . This is small enough that we can explicitly solve cd for A−1 . Fact If ad − bc = 0, then −1 d −b 1 ab = . −c a cd ad − bc
61. 61. Cofactors Remember how we computed determinants: Given A, Cij was (−1)i+j times the determinant of Aij , which was A with the ith row and j column deleted. Let C+ = (Cij ). Then A(C+ ) = |A| I So 1 A−1 = (C+ ) |A| We denote by adj A the matrix (C+ ) , the adjoint matrix of A.
62. 62. Computing inverses with row reduction To ﬁnd the inverse of A, form the augmented matrix A I and row reduce. If the RREF has the form I B then B = A−1 . Otherwise, A is not invertible.
63. 63. Example   0 2 −3 Find the inverse of A = −2 1 2  using row reduction. 201 0 2 −3 1 0 0 ← −    20 1001  −2 1  −2 1 2 0 1 0 ← + − 2 0 1 0 0 2 −3 1 0 0 20 1001 ← −   20 1001 0 1 3 0 1 1  −2 0 2 −3 1 0 0 ← +−   20 10 0 1 0 1 30 1 1 0 0 −9 1 −2 −2 ×−1/9
64. 64. 1 ← − −+  −−  201 0 0 1 ← + − 0 1 3 0 1 0 0 1 −1/9 2/9 2/9 −3 −1  ×1/2  −2/9 2 00 1/9 7/9 0 10 1/3 1/3 1/3  0 1 −1/9 0 2/9 2/9   0 0 1/18 −2/18 1 7/18 0 10 1/3 1/3 1/3  0 1 −1/9 0 2/9 2/9 Notice we get the same matrix as with the adjoint method.
65. 65. Properties of the inverse Theorem (Properties of the Inverse) Let A and B be invertible n × n matrices. Then (a) (A−1 )−1 = A (b) (AB)−1 = B−1 A−1 (c) (A )−1 = (A−1 ) (d) If c is any nonzero number, (cA)−1 = c −1 A−1 .