Linear Algebra Homework and Study Guide

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1 Linear Algebra Homework and Study Guide Phil R. Smith, Ph.D. February 28, 20 Homework Problem Sets Organized by Learning Outcomes Test I: Systems of Linear Equations; Matrices Lesson. Give examples of linear equations using the definition. Generate at least three examples of linear equations on your own and write them down in a list. An equation of n variables x, x 2,..., x n is said to be linear if it is in the form a x + a 2 x a n x n = b where a, a 2,..., a n and b are real number constants. 2. Distinguish linear equations from nonlinear equations. A linear combination of unknowns (or variables) is the weighted sum of (a) Which of the following are linear equations in x, y, and z? the unknowns, each term of which is the product of one real number and i. x 2 3y + 5z = 3 one unknown. For example, 3x + 5y, ii. xyz = 2 2x 3x 2 + 5x 3, and ax + by + cz (with a, b, c real-valued constants) are linear iii. 2x 3z = 2 combinations of {x, y}, {x, x 2, x 3 }, and {x, y, z} respectively. A linear (b) Which of the following are linear equations in x, x 2, and x 3? combination of unknowns set equal to a constant is said to be a linear i. 2x 3 x 2 + 5x 3 = equation. Thus, the following are linear equations: 3x + 5y =, 2x 3x 2 + 5x 3 = 3, and ax + by + cz + d = 0 ii. x = 5x 2 + 7x 3 iii. x + x 2 + 2x 3 = 4 iv. x + 5x 2 + x x 3 = 8 v. x 3/5 + 2x 2 4x 3 = vi. πx + 2x x 3 = 5 /3 (c) If k is a real-valued constant, which of the following are linear equations? i. x x 2 + x 3 = sin k ii. kx k x 2 = 0 iii. 2 k x + 7x 2 x 3 = 0 3. Write solutions sets to linear equations using standard mathematical notation, including parametric notation for linear equations with an infinite number of solutions. (a) Find the solution set of the following linear equations. i. 5x 7y = 4 ii. 3x 5x 2 + 4x 3 = 7 iii. 3x 8x 2 + 2x 3 x 4 + 4x 5 = 0 (b) Find a linear equation in x and y that has the general solution x = 5 + 2t, y = t. The solution set of a linear equation is a list or description of the numbers that make the linear equation true.

2 linear algebra homework and study guide 2 (c) Show that x = s, y = 2 s 5 2 is also a general solution of the equation found in problem 3(b) above. 4. Explain the interconnections between the terms linear combination, linear equation, variable, unknown, and solution set. Create a diagram showing the relationships among the following concepts: linear combination, linear equation, variable, unknown, and solution set. Lesson 2. Give examples of systems of linear equations based on the definition. Generate several examples of linear systems and write them down in a list. Figure : Example concept map. Major concepts are represented by squares, rectangles, circles, or ovals. Connections between concepts are represented with labeled line segments or labeled arrows. 2. Recognize and translate among notations for list, column, and row vectors. Complete the table of equivalent vectors below with the appropriate list, column, or row vectors.

3 linear algebra homework and study guide 3 (a) u = (5, 3, 2) = =. 2 (b) v = = 8 =. 6 [ ] (c) w = = = (d) r = (a, b, c, d, e) = =. ] (e) x = = = [x x 2 x 3 x 4 x 5 x 6 x Recognize and translate among formats for writing linear systems. In the problems below, a linear system has been described as either a system of linear equations, a matrix equation, a column vector equation, or an augmented matrix. Rewrite the each system below in each of the remaining three formats (a) x 9 (b) 2 8 x 2 = x (c) m 2 + n 5 + p 2 = (d) 2y + 3y 2 + 5y 3 + 7y 4 = 5 y + 5y 3 = y + 5y 2 + 2y 3 y 4 = 2 24y 2 + 2y 4 = 5 4. State the definitions of inconsistent and consistent linear systems. Write the definitions in your own words from memory. Be precise. 5. Illustrate using two-dimensional graphs linear systems with no solutions, one solution, and an infinite number of solutions. Draw three 2-D graphs, one for each case. 6. Identify the effects that coefficients have on solutions to linear systems, specifically how they can determine whether the system has no solutions, one solution, or an infinite number of solutions. (a) For which value(s) of the constant k does the system A system of linear equations: { 2x + 3y = 6 x + y = A [ matrix ] [ equation ] [ ] Ax = b: 2 3 x 6 = y A column vector equation xc [ + ] yc 2 [ = ] b: [ ] x + y = An augmented matrix [ A b ] : [ ] x y = 5 2x 2y = k

4 linear algebra homework and study guide 4 have no solutions? Exactly one solution? Infinitely many solutions? Explain your reasoning. (b) Consider the system of equations x + y + 2z = a x + z = b 2x + y + 3z = c Show that for this system to be consistent, the constants a, b, and c must satisfy c = a + b. (c) For which value(s) of the constant a does the system x + 2y 3z = 4 3x y + 5z = 2 4x + y + (a 2 4)z = a + 2 have no solutions? Exactly one solution? Infinitely many solutions? Explain your reasoning. Lesson 3. Write an arbitrary system of equations in general form. An arbitrary system of 3 linear equations in 4 unknowns: Write the system for r linear equations in s unknowns. Let the a x + a 2 x 2 + a 3 x 3 + a 4 x 4 = b constant c be the coefficient of the unknowns and let d be the a 2 x + a 22 x 2 + a 23 x 3 + a 24 x 4 = b 2 constant that follows the = sign. a 3 x + a 32 x 2 + a 33 x 3 + a 34 x 4 = b 3 2. Write an arbitrary augmented matrix in general form. Write the augmented matrix for r linear equations in s unknowns. Let the constant c be the coefficient of the unknowns and let d be the constant that follows the = sign. 3. State the three elementary row operations. Write a description of the row operations in your own words from memory. 4. Review: Solve a 3-equation in 3-unknowns linear system using intermediate algebra methods. Solve the following system using the methods you learned in x y z = intermediate algebra: 3x y + z = x 2y z = 2 5. Gauss-Jordan by hand: Solve a 3-equation in 3-unknowns linear system using elementary row operations and the Gauss-Jordan elimination algorithm. Solve the following system using elementary row operations and x y z = the Gauss-Jordan elimination algorithm: 3x y + z = x 2y z = 2 An arbitrary system of m linear equations in n unknowns: a x + a 2 x a n x n = b a 2 x + a 22 x a 2n x n = b a m x + a m2 x a mn x n = b m An arbitrary augmented matrix of m linear equations in n unknowns: a a 2 a n b a 2 a 22 a 2n b a m a m2 a mn b m

5 linear algebra homework and study guide 5 6. Technology - RREF command: Use technology to solve a 3- equation in 3-unknowns linear system. Solve the following system using a graphing calculator or other technology: x y z = 3x y + z = x 2y z = 2 7. Determine whether a row of an augmented matrix has a leading or not. A zero row is a row of a matrix in Give three examples of row vectors with leading s and three rows which all the entries are zero. A nonzero row is matrix row that without. contains at least one nonzero entry. 8. State the three requirements for an augmented matrix to be considered in row-echelon form (REF). In your own words and from memory, write the three requirements for an REF matrix. 9. State the four requirements for a matrix to be considered in reduced row-echelon form (RREF). In your own words and from memory, write the four requirements for an RREF matrix. 0. Determine whether a matrix is in row-echelon form only (REF only), reduced row-echelon form (RREF), or neither (a) 0 0 (b) 0 0 (c) 0 0 (d) (e) 0 0 (f) (g) 0 (h) (i) (j) (k) The first nonzero element of a matrix row is called a leading coefficient. If the leading coefficient is a, then it is called a leading or pivot. A matrix is said to be in row echelon format (REF) if the first three of the following conditions are met and in reduced row echelon format (RREF) if all four of the following conditions are met: (a) If there are zero rows in the matrix, they are grouped at the bottom of the matrix. (b) If a matrix row is nonzero, then its first nonzero entry is a. (c) After the first, the leading of each nonzero row occurs to the right of the leading entry of the previous row. (d) In each column that contains a leading, all entries in the column above and below the leading are zeros.

6 linear algebra homework and study guide (l) (m) Given the RREF matrix for a system of linear equations, solve (read off) the solutions and write them in standard solution notation. The augmented matrices shown below have reduced by row operations to RREF. Solve each system (a) (b) (c) (d) (e) Compare and contrast equivalent mathematical definitions. Consider the following alternative definition of REF: A matrix is said to be in row echelon form if each of its nonzero rows has more leading zeros than the previous row. Is this definition of row echelon form that same as the one you were given in class? If not, modify this definition so that it is the same. Lesson 4. Apply the Gauss-Jordan elimination algorithm. x + y + z = 2 (a) The linear system 2x 2y z = 2 is being solved with the 3x + y 2z = 2 Gauss-Jordan algorithm. For each step of the algorithm, state the elementary row operation(s) that generated the step. Use codes like R R 3, 2 R 3 R 3, and 2R + R 4 R 4 to represent elementary row operations.

7 linear algebra homework and study guide 7 2 i ii iii iv v vi vii (b) The augmented matrix below is in the middle of the Gauss- Jordan elimination algorithm. Describe the next three row operations. There s no need to actually perform the row operations; just describe them using elementary row operation codes like 2R + R 3 R (c) Solve each system with the Gauss-Jordan elimination algorithm. Document your transformations from one matrix to another with elementary row operation codes like 3 4 R 2 R 2. i. x + x 2 + 2x 3 = 8 x 2x 2 + 3x 3 = 3x 7x 2 + 4x 3 = 0 ii. 2b + 3c = 3a + 6b 3c = 2 6a + 6b + 3c = 5

8 linear algebra homework and study guide 8 iii. 2x 3x 2 = 2 2x + x 2 = 3x + 2x 2 = iv. 5x 2x 2 + 6x 3 = 0 2x + x 2 + 3x 3 = 2. Solve a linear system by learning the appropriate keystrokes on a graphing/symbolic calculator. Solve each system with a graphing/symbolic calculator. Write the augmented matrix that was entered in the calculator, then transcribe the resulting RREF matrix from your calculator screen, followed by a statement of the solution in usual mathematical form. (a) x y + 2z w = 2x + y 2z 2w = 2 x + 2y 4z + w = 3x 3w = 3 [ ] To solve using a TI-89 3 calculator, enter rref([,, ;, -, 3]). Enter the rref function by selecting Matrix from the menu (by pressing 2nd MATH 4), followed by selection of rref from the menu (by pressing 4). Then enter the matrix. Note that entries in each row are separated by commas, and each row is separated by a semicolon. (b) 3x + 2x 2 x 3 = 5 5x + 3x 2 + 2x 3 = 0 3x + x 2 + 3x 3 = 6x 4x 2 + 2x 3 = 30 (c) x 2x 2 + x 3 4x 4 = x + 3x 2 + 7x 3 + 2x 4 = 2 x 2x 2 x 3 6x 4 = 5 3. Generate and solve systems of linear equations from a variety of mathematical contexts. (a) Solve the following system for x, y, and z. x + 2 y 4 z 2 x + 3 y + 8 z x + 9 y 0 z = = 0 = 5 (b) The following points lie on the cubic equation y = ax 3 + bx 2 + cx + d: (0, 0), (, 7), (3, ), (4, 4). Find the coefficients a, b, c, and d and write the equation for the curve. (c) The following points lie on the circle ax 2 + ay 2 + bx + cy + d = 0: ( 2, 7), ( 4, 5), and (4, 3). Find the coefficients a, b, c, and d and write the equation for the circle.

9 linear algebra homework and study guide 9 Lesson 5. Define a trivial solution to a system of linear equations. In mathematics, the word trivial is From memory and in your own words, state the definition of a used to describe a solution or an example of mathematical structure trivial solution to a linear system of equations. that is ridiculously simple and/or immediately obvious. For example, 2. Define a nontrivial solution to a system of linear equations. it takes no virtually no effort to see From memory and in your own words, state the definition of a that x = 0, y = 0 is a solution to nontrivial solution to a linear system of equations 3x + 2y = 0 but more effort is required for nontrivial solutions like x = 2, y = 3. Give an example of a trivial solution. 3 and x = 3, y = 2. Write a trivial solution for a system with 8 unknowns. 4. Give an example of a nontrivial solution. Write a nontrivial solution for a system with 7 unknowns. 5. Give an example of a homogeneous system of linear equations. Write a specific homogeneous system of 3 equations in 4 unknowns. 6. Determine whether a homogeneous system has infinitely many solutions using the Number of Solutions to a Homogeneous System of Equations Theorem. (a) Determine by inspection (i.e., no calculations with paper and pencil, or calculator) which of the following homogeneous systems have nontrivial solutions. i. 2x 3x 2 + 4x 3 x 4 = 0 7x + x 2 8x 3 + 9x 4 = 0 2x + 8x 2 + x 3 x 4 = 0 Number of Solutions to a Homogeneous System of Equations Theorem. A homogeneous system of linear equations with more unknowns than equations has infinitely many solutions. ii. x + 3x 2 x 3 = 0 x 2 8x 3 = 0 4x 3 = 0 iii. a x + a 2 x 2 + a 3 x 3 = 0 a 2 x + a 22 x 2 + a 23 x 3 = 0 iv. 3x 2x 2 = 0 6x 4x 2 = 0 (b) Solve the following homogeneous systems of linear equations by any method. i. 2x + x 2 + 3x 3 = 0 x + 2x 2 = 0 x 2 + x 3 = 0

10 linear algebra homework and study guide 0 ii. 3x + x 2 + x 3 + x 4 = 0 5x x 2 + x 3 x 4 = 0 iii. 2x y 3z = 0 x + 2y 3z = 0 x + y + 4z = 0 iv. v + 3w 2x = 0 2u + v 4w + 3x = 0 2u + 3v + 2w x = 0 4u 3v + 5w 4x = 0 7. Compare and contrast homogeneous and nonhomogeneous systems of linear equations with more unknowns than equations. The Number of Solutions to a Homogeneous System of Equations Theorem only applies only to homogeneous systems of equations. So, although it is true that every homogeneous linear system with more unknowns than equations will have infinitely many solutions, it is not necessarily the case for nonhomogeneous linear systems. Find a nonhomogeneous linear system with more unknowns than equations but does not have infinitely many solutions. Lesson 6. Give the size of a matrix using the standard mathematical convention. Generate specific example matrices of size 2 3, 4 2, and Give examples of square, column, and row matrices. (Note that column matrices are sometimes called column vectors, and row matrices are sometimes called row vectors.) Generate specific examples of a square matrix, a column matrix, and a row matrix. 3. For a specific square matrix, identify the main diagonal and trace of the matrix. 6 3 (a) if A = 2 4 5, then compute tr(a) (b) if B = 4 8, then compute tr(b). 9 2

11 linear algebra homework and study guide 2λ 2 0 (c) Solve the following equation tr(c) = 0 where C = λ 4. Determine if two matrices are equal. Matrix Arithmetic Properties Theorem. If the sizes of the matrices A, B, and C (a) Use the definition of equal matrices to solve the following equa- are such that the stated operations can be performed and k and l are any two real number constants, then tion for a, b, c, and d: [ a b 3d + c ] [ b + c = 2a 4d (b) Are the matrices A and B equal? Why or why not? [ ] [ ] A =, B = Calculate scalar multiples of a matrix, add two matrices, and form linear combinations of matrices (a) Compute 2 3 A if A = [ ] (b) Compute k 2 0 B if B =. 3 [ ] [ ] (c) Compute C + D if C = and D = [ ] [ ] (d) Compute E + F if E = and F = (e) Compute 2G 3H if G = and H = ] (a) Commutative Property of Addition: A + B = B + A (b) Associative Property of Addition A + (B + C) = (A + B) + C (c) Associative Property of Multiplication A(BC) = (AB)C (d) Left Distributive Property: A(B + C) = AB + AC (e) (f) (g) (h) Right Distributive Property: (A + B)C = AC + BC (k + l)a = ka + la k(la) = (kl)a k(ab) = (ka)b = A(kB) Definition of Matrix Subtraction. For any two matrices A and B of the same size, the difference of A and B is defined as: A B = A + ( B) 6. Generate the transpose of a matrix. Transpose Properties Theorem. If the [ ] sizes of the matrices are such that the Give the transpose of the following matrices A =, stated operations can be performed, 0 2 then 2 (a) (A T ) T = A 5 a (b) (A + B) T = A T + B T B = 7, and C = b 9. (c) (ka) T = ka T, where k is any scalar 3 3 c 8.5 (d) (AB) T = B T A T 7. Determine when two matrices may be multiplied. (a) Let A be of size 3 4; B be size 3 4; C, 4 2; D, 3 2; and E, 4 3. Determine which of the following products are defined. For those that are defined, give the size of the resulting matrix. i. AC

12 linear algebra homework and study guide 2 ii. BA iii. CD iv. A T B (b) Prove the following: If the matrix multiplications AB and BA are defined, then AB and BA are both square matrices. 8. If it is possible to multiply two matrices, perform the calculations by hand. [ ] [ ] (a) 2 0 [ ] [ ] x m 4 p (b) 2 q 2 x [ ] 2 (c) y z 8 a 0 (d) b c 9. If it is possible to multiply two matrices, perform the calculations by graphing/symbolic calculator (a) [ ] (b) [ ] (c) Determine the size of a matrix that is the result of some combination of matrix operations. Let A be a p q matrix and let B be an s t matrix. Complete the following statements in terms of the variables p, q, s, and t. (a) AB is defined provided that. (b) A + B 2 is defined provided that. (c) A 3 is defined provided that. (d) If BA is defined, then (BA) T is a(n) (e) If B = B T, then. matrix.

13 linear algebra homework and study guide 3. Generate a matrix from a formula for an arbitrary entry a ij. (a) Find the 4 4 matrix A = a ij such that every arbitrary entry satisfies the formula below: i. a ij = i 2 + j ii. a ij = j i, if i j > iii. a ij =, if i j (b) An identity matrix I is a matrix in which the diagonal entries (top,leftmost corner to bottom, rightmost corner) are all ones and all other matrix entries are zero. Write a formula for a ij that generates all n n identity matrices I. Lesson 7. Explain why matrix multiplication is not commutative. An identity matrix, designated I, is the Explain mathematically why the matrix products AB and BA need matrix that when multiplied by any other matrix returns the matrix. That is, not be equal. AI = IA = A for all A. 2. Describe in symbolic form the following laws of matrix arithmetic: a) commutative law for addition, b) associative law for addition, c) associative law for multiplication, d) left distributive law, and e) right distributive law. List the properties above from memory. The inverse of matrix A, designated A, is the matrix that when multiplied by the original matrix returns the identity. That is, AA = A A = I for all A. 3. Define the zero matrix and describe its associated properties. A matrix A that has an inverse is said Give specific examples of zero matrices at the following sizes: to invertible. A matrix B without an inverse is said to be not invertible or 2 2, 3, 4, and 2 3. noninvertible. There is an alternate terminology that emphasizes the 4. Define the identity matrix and give examples at various sizes. reverse. A matrix without an inverse From memory, state the definition of the identity matrix. Write the is also said to be singular and a matrix 3 3 identity matrix and the 5 5 identity matrix. with an inverse would then be said to be nonsingular. In general, I tend to 5. Define the inverse of a matrix. use use invertible and not invertible to describe matrices, but you should be From memory, state the definition of the inverse of a matrix. familiar with both sets of terminology. 6. Compute the inverse of a 2 2 matrix. The formula for computing the inverse of 2 2 matrix is = [ ] [ ] a b 3 5 c d (a) [ ] 2 d b. [ ] ad bc c a 2 3 (b) 4 4 [ ] 3 0 (c) 0 2 (d) [ ]

14 linear algebra homework and study guide 4 7. Describe the circumstances in which a matrix would not have an inverse. Give all the possible types of 2 2 non-invertible matrices. 8. Use properties of matrix operations, transpose, and inverse to solve matrix equations. [ ] (a) A 2 3 = 5 [ ] (b) (5A) 2 = 3 7 [ ] (c) (7A T ) 5 2 = 3 [ ] (d) (I + 3A) 4 5 = 2 9. Evaluate [ functions ] in which matrices are taken as the inputs. 2 If A =, find the value of f (A) if f (x) = x 2 + 3x Describe how the inverse of a matrix product is related to the individual inverses of each matrix. (a) The Inverse-Product Theorem states that (AB) = B A provided A and B are invertible and of the same size. Extend this result to three matrices: Prove: If A, B, and C are invertible matrices of the same size, then (ABC) = C B A. (b) In the problem below, assume that all matrices are the same size and invertible. Solve for the X matrix. Lesson 8 BCA T XCB T A = AC T. Explain the components of a deductive system of inquiry. Make a list of the components with a brief description of each. In other words, we multiply the corners of the orginal matrix and subtract to find the determinant. Then the inverse is the reciprocal of the determinant times a new matrix in which a and d have exchanged places and we are taking the opposite of the b and c entries. Inverse-Product Theorem. If A and B are invertible matrices of the same size, the product of AB is invertible and the inverse of the product is given by: (AB) = B A. In other words, the inverse of the product is equal to the reverse product of the inverses. Inverse Matrix Properties Theorem. If A is invertible and n is an integer such that n 0, then (a) (b) A is invertible and (A ) = A A n is invertible and (A n ) = A n = (A ) n. (c) ka is invertible for any scalar k = 0 and (ka) = k A 2. Compare and contrast a deductive system of inquiry like mathematics with an inductive system like chemistry or physics. Make a side-by-side chart comparing/contrasting the two systems of inquiry. 3. Describe the basic components of an If-Then proof. (a) Prove: The square of an even number is also even.

15 linear algebra homework and study guide 5 (b) Prove: If A and B are n n matrices, then tr (A + B) = tr (A) + tr (B). 4. Identify symmetric matrices. A square matrix A is said to be symmetric if A T = A. (a) Which of the following matrices are symmetric? [ ] 3 i. 2 [ ] 7 3 ii iii iv v (b) Find all values a, b, and c for which A is symmetric. 2 a 2b + 2c 2a + b + c 3 5 a + c (c) Using as many variables a, b, c,..., z as necessary, write formulas for symmetric 2 2, 3 3, and 4 4 matrices. (d) Let A = [a ij ], B = [b ij ], C = [c ij ], and D = [d ij ] be n n matrices. Determine whether the matrices A, B, C, and D are symmetric. i. a ij = i 2 + j 2 ii. b ij = ij iii. c ij = i 2 j 2 iv. d ij = i j 5. Establish properties of symmetric matrices through mathematical proof. (a) Prove: If A is a square matrix, then A + A T is symmetric. (b) Prove: If B is a square matrix, then BB T is symmetric. (c) Prove: If A is a symmetric matrix, then A 2 is also a symmetric matrix. i. Proof. Assume A is a symmetric matrix. ii. It follows that A = A T.

16 linear algebra homework and study guide 6 iii. In order to demonstrate that A 2 is symmetric, it suffices to show that. iv. (A 2 ) T = (AA) T v. = A T A T vi. = AA vii. = A 2 viii. Therefore, A 2 is symmetric by definition. Q.E.D. (d) Prove: If A T A = A, then A is a symmetric matrix. (e) Prove: If A T A = A, then A = A Establish properties of skew-symmetric matrices through mathematical proof. (a) Prove: If A is an invertible, skew-symmetric matrix, then A is also a skew-symmetric matrix. (b) Prove: If A is skew-symmetric, then so is A T. (c) Prove: If A and B are skew-symmetric matrices, then so is A + B. (d) Prove: If C is a square matrix, then C C T is skew-symmetric. i. Proof. (write the st step of an if-then proof) ii. In order to demonstrate that C C T is skew-symmetric, it suffices to show that (C C T ) T = (C C T ). iii. (C C T ) T = C T (C T ) T iv. = C T C v. = C + C T vi. = (C C T ) vii. Therefore, C C T is skew-symmetric. Q.E.D. 7. Describe how the inverse of the transpose of a matrix is related to the transpose of the inverse of a matrix. Prove: If A is an invertible symmetric matrix, then A is also symmetric. (a) Proof. Assume A is both invertible and symmetric. (b) It follows that A = A T. (c) In order to demonstrate that A is symmetric, it suffices to show that (A ) T = A. (d) (A ) T = (A T ) (e) = A (f) Therefore, A is symmetric by definition of symmetric matrices. Q.E.D. A square matrix A is said to be skew-symmetric if A T = A. Inverse-Transpose Theorem. If A is an invertible matrix, then A T is also invertible and (A T ) = (A ) T.

17 linear algebra homework and study guide 7 Lesson 9. Given an elementary matrix, state the associated elementary row operation. For each elementary matrix below, give the associated row operation (e.g., 2R + R 3 R 3 ) that would be performed if a given matrix was left-multiplied by the elementary matrix (a) E 4 = ( ) 0 0 (b) 2 E 5 3 = (c) E 2 ( ) = [ 0 0 ] 2. Explain how row operations can be represented as multiplications of elementary matrices Consider the matrices A = 2 7 5, B = and C = (a) What is the specific elementary matrix (use one of the following notations: E ij, E i (k), or i E j (k)) that left-multiplies A to produce matrix B? (b) What is the specific elementary matrix (use one of the following notations: E ij, E i (k), or i E j (k)) that left-multiplies B to produce matrix A? (c) What is the specific elementary matrix (use one of the following notations: E ij, E i (k), or i E j (k)) that left-multiplies A to produce matrix C? (d) What is the specific elementary matrix (use one of the following notations: E ij, E i (k), or i E j (k)) that left-multiplies C to produce matrix A? An elementary matrix, E, is simply an identity matrix that has had one elementary row operation applied to it. Left-multiplying a matrix by an elementary matrix is the same as applying the elementary row operation to it. Below is a specialized notation for the three types of elementary matrices: (a) (b) (c) E ij represents an elementary transposition matrix in which row i was swapped with row j, e.g., 0 0 E 23 = 0 0 will cause row and 3 to be swapped. E i (k) represents an elementary diagonal matrix in which the leading in the ith row is multiplied by the constant k. This has the effect of multiplying the ith row by the given scalar. For example, 0 0 E 2 ( 2 ) = has the effect 0 0 of multiplying the 2nd row by 2. ie j (k) represents an elementary row replacement matrix in which row i is multiplied by scalar k and added to 0 2 row j, e.g., 3 E ( 2) = has the effect of multiplying row 3 by 2 and adding the result to row to become the new row. 3. Recognize that an invertible matrix can be written as a product of elementary matrices. Write matrices A and B below as products of elementary matrices. (a) A = [ ]

18 linear algebra homework and study guide (b) B = Find inverses of square matrices [ of higher ] order than 2 2. ] Using row operations to convert A I [I A, find the inverse of the following matrix. 0 A = Find the inverse of square matrices of higher order than 2 2 using a symbolic calculator. Find the inverses of the following matrices: 3 0 (a) A = (b) B = (c) C = Lesson 0. Solve a linear system by first writing it in matrix form, then multiplying both sides by the inverse of the coefficient matrix. (a) Solve the following systems of equations by inverting the coefficient matrix and using the Unique Solution Theorem. i. 4x 3x 2 = 3 2x 5x 2 = 9 Unique Solution Theorem. If A is an invertible square matrix (n n) and b is a column vector of constants (n ), then the system of equations Ax = b has exactly one solution, namely, x = A b. ii. x + 3x 2 + x 3 = 4 2x + 2x 2 + x 3 = 2x + 3x 2 + x 3 = 3 iii. 5x + 3x 2 + 2x 3 = 4 3x + 3x 2 + 2x 3 = 2 x 2 + x 3 = 5

19 linear algebra homework and study guide 9 (b) Solve the following general systems of equations by inverting the coefficient matrix and using Unique Solution Theorem. Then use the resulting formula for x to find the specific solution for a specific b. i. ii. iii. iv. v. 3x + 5x 2 = b x + 2x 2 = b 2 where b = 3x + 5x 2 = b x + 2x 2 = b 2 where b = [ [ b b 2 b b 2 x + 2x 2 + 3x 3 = b 2x + 5x 2 + 5x 3 = b 2 where b = 3x + 5x 2 + 8x 3 = b 3 x + 2x 2 + 3x 3 = b 2x + 5x 2 + 5x 3 = b 2 where b = 3x + 5x 2 + 8x 3 = b 3 x + 2x 2 + 3x 3 = b 2x + 5x 2 + 5x 3 = b 2 where b = 3x + 5x 2 + 8x 3 = b 3 ] [ ] 4 =. 3 ] [ ] 2 =. b b 2 = 3. b 3 4 b 5 b 2 = 0. b 3 0 b b 2 = Solve a matrix equation by using the inverse of a matrix. Solve the following matrix equation for X: X = Apply properties of diagonal matrices to solve various mathematical problems. (a) Determine whether the diagonal matrix below is invertible; if it is, find the inverse by inspection. [ ] 3 0 i ii b 3 iii (b) The matrix products below include at least one diagonal matrix as a factor. Compute the product by inspection i

20 linear algebra homework and study guide 20 [ ] ii iii (c) Find A 2, A 2, and A k by inspection. [ ] 0 i. Let A = ii. Let A =

21 linear algebra homework and study guide 2 Test II: Determinants Lesson. Count the number of inversions in a permutation and classify whether the permutation is even or odd. For the permutations below, give the number of inversions and indicate ( whether the ) permutation ( is odd ) or even. ( ) (i) 3 ( (ii) ) ( (iii) ) 5 ( ) (iv) 4 ( (v) ) (vi) (vii) A permutation is an arrangement of a finite set of numbers. In a permutation, each element of the finite set can be used only one time. So ( 2 3 ), ( 3 2 ), and ( 3 2 ) are examples of permutations of the set {, 2, 3} but ( 2 3 ) and ( ) are not. 2. Find elementary products of a given matrix. Find all the elementary products of matrices B and C. Make sure that, in each elementary products, the factors are ordered with the i index in numerical order (e.g., b b 2 b 3 or c c 2 c 3 c 4 ). An elementary product of a n n matrix A is a product of n entries from b b 2 b 3 A, no two of which come from the same (a) B = row or the same column. b 2 b 22 b 23 b 3 b 32 b 33 c c 2 c 3 c 4 c (b) C = 2 c 22 c 23 c 24 c 3 c 32 c 33 c 34 c 4 c 42 c 43 c Find signed elementary products of a given matrix. Each elementary product can be ordered according to its first index, For matrices B and C in the previous problem, find all of the e.g., a 3 a 22 a 3. The j indices form an signed elementary products. elementary product s permutation, e.g., ( 3 2 ). A signed elementary product is simply the product of 4. Calculate the determinant by using the definition. Use the definition to calculate each of the determinants below. ( ) # of inversions of the permutation and its Specifically, in each case, find all the elementary products, deter- associated elementary product, e.g. ( ) 3 a 3 a 22 a 3. mine their respective signs, and then add the signed products. 3 7 (a) (b) (c) 2 7 a 3 5 (d) 3 a (e) The determinant of a square matrix is the sum of all of its signed elementary products. The determinant of a matrix A is denoted either det (A) or A.

22 linear algebra homework and study guide 22 (f) (g) c c 2 4 c 2 5. Calculate the determinant by using a graphing/symbolic calculator. Use your TI-89 calculator or equivalent technology to check your calculations of the determinants of the matrices in the previous problem. 6. Solve equations involving determinants. [ ] µ (a) Find all values µ for which det (A) = 0 if A =. 4 µ 4 ψ (b) Find all values ψ for which det (B) = 0 if B = 0 ψ ψ 4 (c) Solve for x: x 2 2 x x = x 3 7. Solve linear systems of equations using Cramer s Rule. Cramer s Rule If Ax = b is a system of n linear equations in n unknowns such that det (A) = 0, then the system has (a) the unique solution: (b) (c) 7x 2x 2 = 3 3x + x 2 = 5 4x + 5y = 2 x + y + 2z = 3 x + 5y + 2z = x 4y + z = 6 4x y + 2z = 2x + 2y 3z = 20 x = det (A ) det (A) x 2 = det (A 2) det (A). x n = det (A n) det (A) where A j is the matrix formed when the jth column of A is replaced with the entries of b. (d) x 3x 2 + x 3 = 4 2x x 2 = 2 4x 3x 3 = 0 (e) x 4x 2 + 2x 3 + x 4 = 32 2x x 2 + 7x 3 + 9x 4 = 4 x + x 2 + 3x 3 + x 4 = x 2x 2 + x 3 4x 4 = 4

23 linear algebra homework and study guide 23 Lesson 2. Calculate determinants by inspection. Basic Properties of Determinants (a) Use basic properties of determinants to calculate the determinants below. Don t use a calculator; look instead for patterns related to the basic properties of determinants i ii iii iv v vi vii viii ix x Zero Row/Column Property: If A is a square matrix with a either a row or column of zeros, then det (A) = 0. Proportional Rows Property: If A is a square matrix with two proportional rows or two proportional columns, then det (A) = 0. Matrix Transpose Determinant Theorem. If A is a square matrix, then det (A T ) = det (A). Effects of Elementary Row Operations on the Determinant: (a) Multiplication of matrix row (or column) by a scalar k multiplies the matrix determinant by that scalar k. (b) Switching two rows (or columns) of a matrix changes the sign of its determinant. (c) Adding a scalar multiple of a row (or column) to another row (or column) has no effect on the determinant.

24 linear algebra homework and study guide xi a b c (b) Given that d e f g h i d e f i. g h i a b c 3a 3b 3c ii. d e f 4g 4h 4i a + g b + h c + i iii. d e f g h i 5a 5b 5c iv. d e f g 2d h 2d i 2d = 5, find the following determinants: Definition of Triangular Matrices. A matrix is said to be triangular if all the entries below (or above) the main diagonal are zero. A matrix in which all of the nonzero entries are at or above the main diagonal is said to be an upper triangular matrix; a matrix in which all of the nonzero entries are at or below the main diagonal is said to be lower triangular. A matrix which is simultaneously upper and lower triangular is said to be diagonal. 2. Calculate determinants by reducing the matrix to row-echelon form. (a) Evaluate the matrix determinants below by reducing the matrix to row-echelon form i ii iii iv v Triangular Matrix Determinant Theorem. If A is an upper triangular, lower triangular, or diagonal matrix, then the det (A) is simply the product of the diagonal entries of A.

25 linear algebra homework and study guide 25 (b) Use row reduction to establish the identity: a b c = (b a)(c a)(c b) a 2 b 2 c 2 Lesson 3. Evaluate determinants of scalar multiplications, sums, differences, products, and quotients where possible using properties of the determinant function. (a) Suppose that A and B are 3 3 matrices with det (A) = 5 and det (B) = 2. Evaluate, if possible, the following determinants using the given specifics and your knowledge of properties of the determinant function. i. det (A 3 ) ii. det (B + A) iii. det (2B) iv. det (B T A) v. det (B ) (b) Suppose that C and D are 4 4 matrices with det (C) = 3 and det (D) = 7. Evaluate, if possible, the following determinants using the given specifics and your knowledge of properties of the determinant function. i. det (C T C ) ii. det (2C + 3D) iii. det ( 2 D2 ) iv. det (D T CD) v. det (3C) 2. Determine when a matrix is invertible by taking its determinant. Use the Determinant Test for Invertibility Theorem to determine which of the following matrices is invertible: 0 (a) (b) (c) Determinant of a Product Theorem. If A and B are square matrices of the same size, then det (AB) = det (A) det (B). Determinant Test for Invertibility Theorem A square matrix A is invertible if and only if det (A) = 0.

26 linear algebra homework and study guide 26 (d) Identify solutions to determinant equations by inspection. Matrix Inverse Determinant Theorem. If A is invertible, then (a) Without actually computing the determinant, show that the det (A ) = det (A). equation below is satisfied by x = 0 and x = 2: x 2 x 2 2 = (b) Without actually computing the determinant, show that: b + c c + a b + a a b c = 0 Lesson 4. Represent 2D and 3D vectors in drawings. (a) Sketch the vectors v = (2, 5), v 2 = ( 3, 4), and v 3 = (0, 5) on the same set of xy-coordinate axes. (b) Sketch the vector w = (3,, 5) within a set of xyz-coordinate axes. (c) Sketch the vector w 2 = (2, 4, 6) within a set of x x 2 x 3 - coordinate axes. 2. Explain the effects of multiplying a vector by various scalar values. (a) What is the effect of multiplying the vector u = (, 3, 4) by the following scalars k? (i) k = 2 (ii) k = (iii) k = 2 (iv) k = 0 (v) k = 2 (vi) k = (vii) k = 2 (b) In general, what is the effect of multiplying any vector u by scalars in the following ranges? i. k > ii. k = iii. 0 < k < iv. k = 0 v. < k < 0 vi. k = vii. k <

27 linear algebra homework and study guide Compute sums of vectors graphically and algebraically. Definition of Vector Addition. Two vectors u = (u, u 2 ) and v = (v, v 2 ) can be added algebraically by adding Let u = (, 4) and v = (2, 3). (a) Add the vectors u and v using the Triangle Law of Vector Addition. (b) Add the vectors u and v using the Parallelogram Law of Vector Addition. (c) Add the vectors u and v algebraically by adding their components. 4. Find linear combinations of vectors. [ ] [ ] 7 2 (a) Calculate av + bw if a = 3, b = 2, v = and w =. 2 [ ] (b) Calculate av + bw if a =, b = 4, v = 2 2 and [ ] w = 3 5. (c) Calculate av + bw if a =.2, b = 0, v = (, 2, 5) and w = (4, 0, 52). (d) Find scalars k and l such that kv + lw = b where v = (2, ), w = (, 2), and b = (, 0) (e) Let u = (2, 0, 4) and v = (, 3, 6). Find values of k and l such that ku + lv = (5, 9, 4). (f) Find scalars c, c 2, and c 3 such that c u + c 2 v + c 3 w = b where u = ( 3,, 2), v = (4, 0, 8), w = (6,, 4), and b = (2, 0, 4) 5. Describe the geometry of a set of linear combinations of vectors in IR 2 and in IR 3. Describe the locus of points contained in the following sets. (a) {k(, ) : k IR} That is, all the scalar multiples of the vector u = (, ) where the scalar ranges over the set of real numbers. (b) {a(, 0) : a Z} That is, all the scalar multiples of the vector u = (, 0) where the scalar ranges over the set of integers. (c) {c(, 0) : c 0, c IR} (d) {a(, 0) + b(0, ) : a Z and b IR} (e) {c(, 0) + d(0, ) : c, d IR} (f) {e(,, 0) + f (0,, ) : e, f IR} their respective components: u + v = (u, u 2 ) + (v, v 2 ) = (u + v, u 2 + v 2 ). Triangle Law: If two vectors are represented by two sides of triangle in sequence (i.e, the tail of the second vector begins at the arrowhead of the first), then the third side of the triangle, in the opposite direction of the sequence, represents the sum (or resultant) of the two vectors. Parallelogram Law: If two vectors are represented by two adjacent sides of a parallelogram, then the diagonal of the parallelogram through the common point represents the sum of the two vectors. Definition of Vector Inverse. A vector x has an additive inverse in IR n if there exists a vector y such that x + y = 0 = y + x. The inverse vector y is denoted x. Furthermore if x = (x, x 2 ) is any vector in IR 2, then x = ( x, x 2 ). Definition of Vector Subtraction. Subtracting a vector is the same as adding its inverse: x y = x + ( y) A locus is a set of points that contains all the points, and only the points, that satisfy the condition, or conditions, required to describe a geometric figure. For example, the locus of points in a plane that are all the same distance 3 from a fixed point would be a circle of radius 3.

28 linear algebra homework and study guide Calculate the Euclidean inner (dot) product and outer product of two vectors. The inner product of u and v: u v = u T v Recall that vectors can be written as column vectors. This allows us to define two types of vector multiplication, an inner product and an outer product. Consider two vectors u = (, 2, 3) and v = (5, 5, 5). To find the inner product, we first view both vectors 5 as 3 column vectors u = 2 and v = 5. It s not possible 3 5 to multiply two 3 matrices. However, taking the transpose of either u or v would allow us to use regular matrix multiplication to combine the two vectors. Note that, for u T v, we are multiplying a 3 matrix times a 3 matrix to produce a matrix, or scalar; and, for uv T, we are multiplying a 3 column matrix by a 3 row matrix to produce a 3 3 square matrix. The vector product u T v is called the Euclidean inner product (or dot product) of u and v and is represented u v. The vector product uv T is called the outer product of u and v and is represented u v. For the vectors u and v below, calculate the dot product u v and the outer product u v. (a) u = (2, ) and v = (3, 7) (b) u = (3,, 6) and v = (2, 5, 3) (c) u = (u, u 2, u 3 ) and v = (v, v 2, v 3 ) = [ 2 3 ] = ()(5) + (2)(5) + (3)(5) = 30 The outer product of u and v: u v = uv T = ] 3 ()(5) ()(5) ()(5) = (2)(5) (2)(5) (2)(5) (3)(5) (3)(5) (3)(5) = Calculate the length of a vector. Definition of Magnitude of Vector: The magnitude, or length, of a vector v is given by v = v v. Find the length of the following vectors: (a) u = (3, ) (b) w = (, 2, 5) (c) x = ( 4, 3) (d) v = (7, 4, 5) 8. Derive relationships about vectors when their lengths are known. (a) If u = 3 and v = 7, what are the largest and smallest possible values for u v? (b) Let w = (2,, 5). Find all scalars k such that kw = 4. (c) Prove: Unit Vector Theorem. If w is any nonzero vector, then ŵ = w is a unit vector. w (d) Use the result above to find a unit vector that lies in the same direction as w = (2, 5). (e) Use the result above to find a unit vector that lies in the same direction as w = ( 2, 6, 3). Magnitude of a Scalar-Vector Product Theorem. kv = k v. Definition of Unit Vector. A vector u is said to be a unit vector if u =. Sometimes, a hat-notation is used to indicate a unit vector in the same direction as a vector. For example, x would represent a unit vector in the same direction as x.

29 linear algebra homework and study guide Determine whether any two vectors u and v are orthogonal, or whether they have an included angle that is acute or obtuse. (a) Find the angle θ between vectors u and v. i. u = (2, ) and v = (3, 7) ii. u = (3,, 6) and v = (2, 5, 3) (b) Without calculating the exact value of the angle, determine whether the vectors u and v make an acute angle, make an obtuse angle, or are orthogonal. i. u = (6,, 4) and v = (2, 0, 3) ii. u = (0, 0, ) and v = (,, ) iii. u = ( 6, 0, 4) and v = (3,, 6) iv. u = (2, 4, 8) and v = (5, 3, 7) (c) Let a = (2, k) and b = (3, 5). Find k such that i. a and b are parallel. ii. a and b are orthogonal. iii. the angle between a and b is π 3. Angle Between Vectors Theorem. If u and v are vectors in either IR 2 or IR 3, then u v = u v cos (θ). Orthogonal Vectors Theorem. If u and v are vectors in either IR 2 or IR 3, then u v if and only if u v = 0. Acute/Obtuse Vectors Theorem. If u and v are vectors in either IR 2 or IR 3 and θ is the angle between them, then θ is acute if and only if u v > 0 and θ is obtuse if and only if u v < 0. iv. the angle between a and b is π Calculate the projection of one vector onto another for 2D and 3D vectors. Calculate the following projections: (a) proj a u where u = (6, 3) and a = (3, 9). (b) proj b w where w = (, 2) and b = ( 2, 3). (c) proj y x where x = (3,, 7) and y = (, 0, 5). (d) proj x y where y = (, 0, 5) and x = (3,, 7).. Given a nonzero reference vector a and an arbitrary vector x, rewrite the vector x as the sum of two vectors, one parallel to a and one orthogonal to a. Find x and x such that x = x + x for the given vectors x and a below: (a) x = (2, 5) and a = (3, ). (b) x = (2, 3, ) and a = (,, ). (c) x = (3, 4, 5) and a = (, 2, 3). 2. Apply the concept of projection to find the distance from a point to a line. (a) Find the distance between the point (2, ) and the line 4x + 2y + 7 = 0. (b) Find the distance between the point (s, t) and the line ax + by + c = 0. Suppose that x and a are two vectors in either IR 2 or IR 3 and a = 0. Then vector x can be written as the resultant (sum) of two vectors x = x + x where x is a vector that is parallel to a and x is a vector that is perpendicular to a. Note that because x is parallel to the vector a, it can also be described as "lying on" a. For this reason, we also call x the projection of x onto a and write x = proj a x. It can be shown that x = x + x where x = proj a x = x a a a a = x a a 2 a and x = x x = x proj a x = x x a a 2 a Applying the Magnitude of a Scalar- Vector Product Theorem to the projection of x onto a, we get proj a x = x a a 2 a x a x a = a =. a 2 a

30 linear algebra homework and study guide Establish properties of the dot product via mathematical proof. The properties in this problem are true in both IR 2 and IR 3. Once you ve proved (a) Prove: Symmetry Property. If u IR 2 and v IR 2, the property in IR 2 (or IR 3 ), it is easy to then u v = v u. write an analogous proof for the IR 3 (or IR 2 ) case. (b) Prove: Additivity Property. If u IR 3, v IR 3, and w IR 3, then (u + v) w = u w + v w. (c) Prove: Homogeneity Property. If k is a real number scalar, u IR 3, and v IR 3, then (ku) v = k(u v). (d) Prove: Positivity Property. If v IR 2, then i. v v 0 ii. v v = 0 if and only if v = 0. Lesson 5. Calculate the cross product of two vectors. If u = (u, u 2, u 3 ) and v = (v, v 2, v 3 ) Compute the vector cross products below by using the definition. are vectors in IR 3, then the cross product u v is defined as (a) v w where v = (3, 2, ) and w = (2, 5, 4). ( u u v = 2 u 3 v 2 v 3, u u 3 v v 3, u u 2 v v (b) w v where v = (3, 2, ) and w = (2, 5, 4). 2 (c) What is the relationship between v w and w v? 2. Compare and contrast the dot product of two vectors with the cross product of two vectors. Make a table that lists similarities and differences between the dot product and the cross product. ) 3. Given two vectors, find a third vector that is orthogonal to both. The cross product of two vectors is orthogonal to each of its component vectors: u v u and u v v Find a vector w that is orthogonal to the vectors u and v. Then verify orthogonality by showing that u w = 0 and v w = 0. (a) u = (3,, 4) and v = (2,, 3). (b) u = ( 2, 0, 5) and v = (,, 7). 4. Find the area of a parallelogram spanned by two vectors. Area(parallelogram spanned by u & v) = u v (a) Find the area of the parallelogram spanned by u = (,, 2) and v = (3,, 4). (b) Find the area of the parallelogram determined by the four points P(, 2), Q(2, ), R(6, ), and S(3, 4). 5. Find the area of a triangle spanned by two vectors. Area(triangle spanned by u & v) = (a) Find the area of the triangle spanned by u = (3,, 3) and 2 u v v = (2, 2, 2). (b) Find the area of the triangle determined by the three points A(0, 4, ), B(6, 2, 5), and C(0, 0, 0)

31 linear algebra homework and study guide 3 6. Given a fixed point and a vector, find the equation of the line that passes through the fixed point and is orthogonal to the vector. Find the equation of a line ax + by + c = 0 that passes through P and is orthogonal to n. (a) P(3, ) and n = (5, ) (b) P( 2, 5) and n = (3, 4) 7. Given a fixed point and a vector, find the equation of the plane that passes through the fixed point and is orthogonal to the vector. Find the equation of a plane ax + by + cz + d = 0 that passes through P and is orthogonal to n. (a) P(, 3, 2) and n = ( 2,, ) (b) P(3, 0, 0) and n = (0, 0, ) (c) P(0, 0, 0) and n = (3, 2, ) Equation for a Line or a Plane Given a Fixed Point and a Normal Vector: In 2-space, let n = (a, b); let P be a fixed point (x 0, y 0 ); and X be an arbitrary point (x, y). In 3-space, let n = (a, b, c); let P be a fixed point (x 0, y 0, z 0 ); and let X be an arbitrary point (x, y, z). Then the equation of the line/plane is given by: n PX = 0 { ax + by + c = 0 (a line in IR 2 ) ax + by + cz + d = 0 (a plane in IR 3 ) 8. Find the equation of the plane that is determined by three given points. (a) P(5, 3, 4), Q(2,, 7), and R(, 4, 5) (b) P( 2, 0, ), Q(, 2, 3), and R(3,, 2) 9. Given a fixed point and a direction vector, find parametric equations for a line in 2- and 3-space. (a) Find the parametric equations x = qt + x 0, y = rt + y 0 of a line that passes through P and is parallel to v = (q, r). i. P( 2, 3) and v = (3, ) ii. P(, 2) and v = ( 5, ) iii. Find parametric equations for the line passing through points P(3, 5) and Q(, 8). (b) Find the parametric equations x = qt + x 0, y = rt + y 0, z = st + z 0 of a line that passes through P and is parallel to v = (q, r, s). i. P( 2, 3, ) and v = (3,, 2) ii. P(2, 2, 5) and v = (, 0, ) iii. Find parametric equations for the line passing through points P(4, 2, 3) and Q(2, 7, 4). 0. Analyze geometric relationships between lines and planes by recognizing normal vectors implicit in equations of the form ax + by + c = 0 and ax + by + cz + d = 0 and recognizing direction vectors for lines in sets of parametric equations like x = qt + x 0, y = rt + y 0 and x = qt + x 0, y = rt + y 0, z = st + z 0 Parametric Equations for a Line Given a Fixed Point and a Direction Vector: In 2-space, let v = (q, r) be the direction vector for the line; let P be a fixed point (x 0, y 0 ) on the line; and X be any point (x, y) on the line. In 3-space, let v = (q, r, s) be the direction vector; let P be a fixed point (x 0, y 0, z 0 ) on the line; and let X be any point (x, y, z) on the line. Then parametric equations of the line/plane can be derived from the vector equation: X = vt + P (x, y) = (q, r)t + (x 0, y 0 ) {(x, y) = (qt + x 0, rt + y 0 ) x = qt + x 0 y = rt + y 0 or X = vt + P (x, y, z) = (q, r, s)t + (x 0, y 0, z 0 ) (x, y, z) = (qt + x 0, rt + y 0, st + z 0 ) x = qt + x 0 y = rt + y 0 z = st + z 0 (a) Determine whether the pairs of planes below are parallel.