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Unformatted text preview: Math 340L, Section 57335 —— Fall 2003 —— lst Midterm
October 1, 2003 Instructor: Manuel Portilheiro Your Name: Question 1: Question 2: l :2;
Question 3: MLE____”__
Question 4: ”L“ Total: Instructions 1. No books, notes or calculators of any kind are allowed.
2. There are four (4) questions in this exam. Answer all questions.
3. Show all your work, and justify your answers. 4. This exam is worth 25% of your course grade. Question 1: Consider the following system of equations equations. IE1 CEQ+ [173' 3342*1
2$1—2$2+4$3 =2
4$1W4$2+5$33$4=~2 (a) Write the augmented matrix associated with this system.
(1)) Find the reduced echelon form of the matrix in (a).
(C) What are the free and basic variables of the system? Question 2: Let A be the matrix A! 2
y;——10—2
A20 05:1?1
0 0 0 0 (a) Does the matrix equation AX : b have a solution for any b E R3?
Explain.
(b) What are the pivot positions of A?
((1) Describe the solution set of the matrix equation Ax = b in para—
3
metric vector form when b = 2
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X W Question 3: Consider the following matrices: 1 ~2 1 —17 4 —1
A: 4 7 3 ,0: —14 3 —1
_2 6 —5 10 2 —1 (a) Compute the product AC. What can you say about C? (b) Are the columns of A linearly dependent or linearly independent?
(0) Without doing any computations, say how many solutions can there
exist for the matrix equation AX 2 b. Question 4: Let B be the matrix B 2 OTC, Where C is the matrix from the previous problem. (a) How can you express B“1 in terms of A? (b) Does the equation BX = 0 have a nontrivial solution?
1 (c) Find the solution of BX = b, Where b = ...
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