MATH 1553, C.J. JANKOWSKI MIDTERM 1

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1 MATH 155, C.J. JANKOWSKI MIDTERM 1 Name Section Please read all instructions carefully before beginning. You have 5 minutes to complete this exam. There are no aids of any kind (calculators, notes, text, etc.) allowed. Please show your work unless specified otherwise. A correct answer without appropriate work may be given little or no credit. You may cite any theorem proved in class or in the sections we covered in the text. Good luck!

2 Scoring Page Please do not write on this page Total

3 Problem 1. [Parts a) through f) are worth 2 points each] a) Compute b) If A is a 2 matrix with 2 pivots, then the set of solutions to Ax = is a: (circle one answer) point line 2-plane -plane in: (circle one answer) R R 2 R. True or false. Circle T if the statement is always true, and circle F otherwise. You do not need to justify your answer. c) T F If a system of linear equations has more variables than equations, then the system must have infinitely many solutions. d) T F If A is an m n matrix and A has a pivot in every column, then the equation Ax = b has a solution for each b in R m. e) T F The three vectors 1, 1 1, and 1 span R. 1 f) T F If u, v, and w are nonzero vectors in R 2, then Span{u, v, w} is R 2. a) = = b) Line in R. Since there are 2 pivots but columns, one column will not have a pivot, so Ax = will have exactly one free variable. The number of entries in x must match the number of columns of A (namely, ), so each solution x is in R. c) False. The system can be inconsistent. For example: x + y + z = 5, x + y + z = 2. 1 d) False. For example, if A = 1, then Ax = is inconsistent. 1 e) True. The three vectors form a matrix with a pivot in every row. 1 f) False. Take v 1 = v 2 = v =. Many other counterexamples possible.

4 Problem 2. [1 points] Johnny Rico believes that the secret to the universe can be found in the system of two linear equations in x and y given by where h is a real number. x y = h x + hy = 4 a) Find all values of h (if any) which make the system inconsistent. Briefly justify your answer. b) Find all values of h (if any) which make the system have a unique solution. Briefly justify your answer. Represent the system with an augmented matrix and row-reduce: 1 1 h R2 R h. h 4 h + 4 h 1 1 a) If h = then the matrix is, which has a pivot in the rightmost column and is therefore 1 inconsistent. b) If h, then the matrix has a pivot in each row to the left of the augment: 1 1 h. The right column is not a pivot column, so the system is h+ 4 h consistent. The left side has a pivot in each column, so the solution is unique.

5 Problem. [11 points] a) Solve the system of equations by putting an augmented matrix into reduced row echelon form. Clearly indicate which variables (if any) are free variables. b) Write the set of solutions to x 1 + 2x 2 + 2x x 4 = 4 2x 1 + 4x 2 + x 2x 4 = 1 x 1 2x 2 x + x 4 = 1 x 1 + 2x 2 + 2x x 4 = 2x 1 + 4x 2 + x 2x 4 = x 1 2x 2 x + x 4 = in parametric vector form. a) R 2 =R 2 2R 1 R =R +R 1 R =R +R 2 R 1 =R 1 2R Therefore, x 2 and x 4 are free, and we have: R 2 R x 1 = 2 2x 2 + x 4 x 2 = x 2 x = x 4 = x b) If we had written the solution to part (a) in parametric vector form, it would be: x 1 2 2x 2 + x x 2 x x = 2 1 = + x 2 + x 4. x 4 x 4 1 The equation in (b) is just the corresponding homogeneous equation, which is a translate of the above plane which includes the origin. x x 2 1 x = x 2 + x 4 (x 2, x 4 real). x 4 1

6 Problem 4. [1 points] The diagram below represents the temperature at points along wires, in celcius. T 1 T T 2 9 Let T 1, T 2, T be the temperatures at the interior points. Assume the temperature at each interior point is the average of the temperatures of the three adjacent points. a) Write a system of three linear equations whose solution would give the temperatures T 1, T 2, and T. Do not solve it. b) Write the system as a vector equation. Do not solve it. c) Write a matrix equation Ax = b that represents this system. Specify every entry of A, x, and b. Do not solve it. 6 a) The left side system below or right-side system below are both fine. T 1 = T 2 + T +, or T 1 T 2 T =. T 2 = T 1 + T + 6, or T 1 + T 2 T = 6. T = T 1 + T 2 + 9, or T 1 T 2 + T = 9. b) T c) 1 + T T1 T 2 = T 1 + T = 6. 9

7 Problem 5. [6 points] Write an augmented matrix corresponding to a system of two linear equations in 4 three variables x 1, x 2, x, whose solution set is the span of 1. Briefly justify your answer. This problem is familiar territory, except that here, we are asked to come up with a system with the prescribed span, rather than being handed a system and discovering the span. Since the span of any vector includes the origin, the zero vector is a solution, so the system is homogeneous. 4 Note that the span of 1 is all vectors of the form t x1 4 1 where t is real. It consists of all x 2 so that x 1 = 4x 2, x 2 = x 2, x =. x The equation x 1 = 4x 2 gives x 1 +4x 2 =, so one line in the matrix can be 1 4. The equation x = translates to 1. Note that this leaves x 2 free, as desired. This gives us the augmented matrix (Multiple examples are possible) Let s check: the system has one free variable x 2. The first line says x 1 + 4x 2 =, so x 1 = 4x 2. The second line says x =. 4x2 4 Therefore, the general solution is x = x 2 = x 2 1 where x 2 is real. 4 In other words, the solution set is the span of 1.

8 The system of equations is x 1 + 4x 2 = x =.

9 [Scratch work]

MATH 1553, SPRING 2018 SAMPLE MIDTERM 1: THROUGH SECTION 1.5

MATH 1553, SPRING 2018 SAMPLE MIDTERM 1: THROUGH SECTION 1.5 MATH 553, SPRING 28 SAMPLE MIDTERM : THROUGH SECTION 5 Name Section Please read all instructions carefully before beginning You have 5 minutes to complete this exam There are no aids of any kind (calculators,