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Linear Algebra Midterm Exam

This document provides instructions for a midterm exam in a Linear Algebra course. The exam contains 5 questions worth a total of 50 marks. Students have 75 minutes to complete the exam and must show all of their working on the exam paper. The exam is worth 20% of the student's overall grade. Question 1 involves finding the intersection of three planes and determining an additional equation. Question 2 involves finding the reduced row echelon form of a matrix and determining properties of the solution set. Question 3 involves factorizing a matrix and determining subspace bases. Question 4 involves determining subspace bases and sketching the fundamental subspaces. Question 5 involves determining subspace bases for given conditions.

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Abiha Ejaz
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145 views12 pages

Linear Algebra Midterm Exam

This document provides instructions for a midterm exam in a Linear Algebra course. The exam contains 5 questions worth a total of 50 marks. Students have 75 minutes to complete the exam and must show all of their working on the exam paper. The exam is worth 20% of the student's overall grade. Question 1 involves finding the intersection of three planes and determining an additional equation. Question 2 involves finding the reduced row echelon form of a matrix and determining properties of the solution set. Question 3 involves factorizing a matrix and determining subspace bases. Question 4 involves determining subspace bases and sketching the fundamental subspaces. Question 5 involves determining subspace bases for given conditions.

Uploaded by

Abiha Ejaz
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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MidTerm Exam

Course: Linear Algebra (L1)


Instructor: Musabbir Abdul Majeed
Time: 75 minutes (First 5 minutes are for reading)
Date: March 6, 2019

ID (just the digits):

Instructions (Yes reading them is required)

1. Do all the questions. There are 5 questions in this exam.


2. Write your answers on the question paper. All working should be shown on the exam
paper. If you want extra paper for your working, please ask
3. The weightage of the exam is 20%

4. You must not start answering the questions before 15:55


5. It is mandatory that you use the given empty space first for each question. If extra space
is required, continue your working at the back of the booklet, and make sure that you
mention the page number near the question
6. Mark distribution is as follows:

Marks
Q1 Q2 Q3 Q4 Q5 Total
10 10 12 10 8 50

1
Questions

1. You are given three equations of four dimensional planes t = 0, z = 0 and x + y + z + t = 1

(a) Write the given problem in form A~x = ~b (3 marks)

(b) Find any two points on the line of intersection of the three planes (5 marks)

(c) Find a fourth equation which will give a unique solution for this system (2 marks)

2
3
2. A~x = ~b can be converted to a reduced row echelon form R~x = d~ using row operations. You are
told that the complete solution of one such system is
     
4 2 5
~x = 0 + c1 1 + c2 0
0 0 1

(a) What is the 3 × 3 reduced row echelon matrix R? (3 marks)


~ (2 marks)
(b) What is d?

(c) What is the dimension of the N (A)? (1 marks)

(d) For the given system A~x = ~b, are we guaranteed to have a solution for every possible ~b?
Give your reasons. (4 marks)

4
5
3. Let A be the matrix  
1 2 1 2
2 1 2 2
1 1 1 1

(a) Find a factorisation A = LU, where L is the lower triangular matrix and U is in echelon
form (4 marks)

(b) What is the rank of A (1 marks)

(c) What are the bases of C(A) and C(U), and C(R), where R is the reduced row echelon
form? (3 marks)

(d) For which right hand sides the system AT ~y is solvable? (4 marks)

6
7
4. You are given the following matrix.
 
1 0 1
2 1 2
A=
0

2 0
1 0 2

(a) Find the dimension and the basis for the four fundamental subspaces (column space, row
space, nullspace and left nullspace) for A (5 marks)

(b) Sketch the big picture of the four fundamental subspaces. Your sketch should include basis
for each of the subspaces, and the area of each rectangle should be proportionate to the
size of the space (5 marks)

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9
5. Find the basis for each of the following subspaces, if they form a subspace or give reasons
otherwise

(a) All 2 × 2 matrices with trace equals to 4.  The trace of a 2 × 2 matrix is defined as
1 2
tr(A) = a11 + a22 . For example if A = , then the trace of A = tr(A) = 5 (5 marks)
3 4

d2 y
(b) Quadratic functions which satisfy dx 2 = 0. Sketch the bases (3 marks)
x=0

10
11
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