MA122 Introductory Linear Algebra (Fall 2022)
Review Guide for the Final Examination
Review Topics: Definitions, theorems, examples, algorithms that are covered in Chapters 4, 5, 6 and 7 of the
textbook.
Note: This review guide should be used in combination with the Review Guide for the Midterm Test, which
contains information on material from the first three chapters of the textbook.
The True-False type problems at the beginning of every section also test the understanding of the important
concepts, theorems and examples. There will be similar questions on the final exam.
You are recommended to organize your review for the final exam by going through the chapter learning outcomes
guides and solving the related exercise problems to make sure that you know the basic concepts and the techniques.
Once you are skillful in working with the problems based on individual learning outcomes, you must consider more
comprehensive problems that combine two or more learning outcomes. Some sample comprehensive problems are
given below.You are recommended to start your review from Chapter 4 and onwards, go back to review earlier
chapters if you difficulty in apply the earlier concepts and methods.
Samples of comprehensive problems which demonstrate some patterns of the final exam
1. Let L : R3 → R3 be a linear operator such that
L(x1 , x2 , x3 ) = (x2 + x3 , x1 + x3 , x1 + x2 ).
(a) Find the standard matrix A = [L].
(b) Show that A is invertible and find A−1 .
(c) Find the inverse operator L−1 (x1 , x2 , x3 ).
(d) Find the standard matrix of the linear operator T = 3L2 − 4L−1 .
(e) Evaluate T (1, 0, −1).
2
2
−
2. (a) Find the standard
matrix of the orthogonal projection linear operator proj→
v : R → R , where the vector
1
→
−
v =
.
−1
(b) Find the standard matrix of the linear operator Rθ : R2 → R2 , the counterclockwise rotation through
2π
radians. (Use the exact values of trigonometric functions.)
angle
3
2π
(c) The linear operator L : R2 → R2 rotates a vector counterclockwise through angle
radians and then
3
projects the resulting vector on the line x1 = −x2 . Determine the standard matrix of L using your answers
in parts (a) and (b).
−1
−
(d) Find the image of the vector →
v =
under L defined in part (c).
1
(e) Find the nullity of L defined in part (c).





1
0
1





0
1
−
−
−
 →

 →
 1
3. Given vectors →
v1=
 2 , v 2 =  0 , v 3 =  2
−1
2
1



1

 2 
−
4
 and →

v4=

 3  in R .
4
−
−
−
−
(a) Determine whether the vectors →
v 1, →
v 2, →
v 3 and →
v 4 are linearly dependent.
→
−
→
−
−
−
(b) Find a basis for the subspace W = Span( v 1 , v 2 , →
v3 ,→
v 4 ) of R4 , and determine dim(W ).
 
1
 1 
−
−
−
−

(c) Determine whether the vector  
lies in Span(→
v 1, →
v 2, →
v3 ,→
v 4 ).
2 
0
(d) Find equations that the components of a vector ⃗b in R4 must satisfy to be in the subspace W of R4 .

2
4. Let A = 1
0
1
2
0

−1
1
3
(a) Find the characteristic polynomial of A.
(b) Find all eigenvalues of A and their algebraic multiplicities.
(c) For each eigenvalue of A found in part (b), determine a basis for the corresponding eigenspace, and
determine its geometric multiplicity.
(d) Determine if the matrix A is diagonalizable using the results in parts (b) and (c).
(e) Find a 3 × 3 diagonal matrix D and a 3 × 3 invertible matrix P such that P D = AP .
(f) Use the result of part (e) to compute A122 .
5. Let L : R3 → R3 be a linear operator with eigenvectors
 
 


1
0
2
→
−
−
−
u = 2 ,→
v =  1  and →
w = 0 
0
2
1
corresponding to eigenvalues λ1 = 1, λ2 = 0 and λ3 = −1 of the standard matrix A = [L] of L respectively.
(a) Find the characteristic polynomial of A (without actually finding A).
(b) Find the matrix A.
(c) Find L(x1 , x2 , x3 ).
6. In the linear system


x − 2y + z = 2
3x + (a − 1)z = b + 1


2x − y + az = b
determine all values of a and b for which the system has
(a) no solution;
(b) a unique solution;
(c) infinitely many solutions.
7. Let A and B be n × n matrices satisfying
A2 + AB − 2B = O
Suppose that 2 is not an eigenvalue of A. Prove that A is singular (not invertible) if and only if B is singular.
You may need to refresh your knowledge and skills learned in the study of the first three chapters before the
midterm test because the final exam is comprehensive and you must use some of them to understand and solve the
problems in later chapters, and/or interpret the solutions of some questions. To help you plan, the following learning
outcome statements summarize the most important concepts in the first three chapters that you may need to use in
the final exam.
• Perform operations on vectors in Rn : addition, subtraction, scalar multiplication, dot product, and work with
vector equations of lines and planes in Rn .
• Transform a matrix to a row echelon form (REF) and to its unique reduced row echelon form (RREF).
• Solve a system of linear equation and display its general solution in standard format as a vector in Rn given by
(a sum of) scalar product(s) of fixed vector(s) with parameter(s) corresponding to the free variable(s), if any,
of the system.
• Perform operations on matrices: addition, subtraction, multiplication, scalar multiplication, transpose and
combinations of these computations.
• Find the inverse of a square matrix using elementary row operations.
• Express linear systems using abstract matrix symbols and use this interpretation to write short proofs about
the solutions of linear systems.
• Apply the equivalent statements for a square matrix to be invertible in computations and proofs.