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Lecture 35

# MATH 240 Lecture Notes - Lecture 35: Linear Combination, Augmented Matrix, Elementary MatrixExam

Department
Mathematics
Course Code
MATH 240
Professor
Michael Monagan
Lecture
35

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MATH 240 Final Exam
April 16th, 2016.
Instructor: Michael Monagan.
Time allowed is 3 hours. There are 16 questions, one per page. Attempt all questions.
The total number of marks is 120. Put your answers in the space provided.
If you need more space, please use the back of a previous page.
Calculators and other electronic devices are not permitted.
A one page sheet of handwritten notes (on both sides) is allowed.
Note, the symbol Rdenotes the set of real numbers.
Full Name: ........................................................................
Student ID number: ...............................................
Signature: ........................................................................
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Question 1 (10 marks)
Answer true (T) if the following statements are always true and false (F) otherwise. You
will receive one mark for each correct answer, no mark for no answer and lose one mark for
In the following u, v, w are vectors in Rnand A, B, C are matrices in Rn×n.
( ) If ||u|| = 0 then uis the zero vector.
( ) u+v=v+u.
( ) u·(v+w) = v·u+w·u
( ) If u·v= 0 and v·w= 0 then u·w= 0.
( ) A(BC) = (AB)C.
( ) If A2is invertible then Ais invertible.
( ) If Aand Bare invertible then (AB)1=A1B1.
( ) Every elementary row operation is reversible.
( ) A linear system with more equations than unknowns has no solutions.
( ) The columns of any 4 ×5 matrix are linearly dependent.
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Question 2 (8 marks)
Consider the following linear system in R3.
2x+y3z= 0
6x+ 3 y8z= 0
2xy+ 5 z=4
Write the linear system in the form Ax =b.
Now solve Ax =bby row reducing the augmented matrix [A|b] to REDUCED row Echelon
form. Identify the solution(s). Show your working.
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