MATH2076-001, Spring 2022
Sample questions
Sample questions (Some relevant concepts/formulas are recalled in “Note” after each question.)
1. Consider the following linear system
Ax = b
with
A=
1
3
3
9
4
7
,b =
7
6
.
(a) Find the reduced row echelon form of the augmented matrix of the above linear system.
(b) Find the pivot positions of the coefficient matrix A.
(c) How many solutions does the linear system have?
Hint: you should be able to answer the two questions above without solving completely the linear
system.
2. Find the inverse of the following

0
A= 1
4
1
0
−3

2
3 .
8
3. Express each of the following 2 transformations of a 3-dimensional vector x as a matrix transformation. That is, each transformation, say denoted by T (x) for x ∈ R3 , is in fact a linear
transformation, and the goal is to write T (x) = M x for some explicit matrix M .
(Note that the transformation can also be extended to 3 × n matrices A = (a1 | · · · | an ) by
T A = T (a1 | · · · | an ) = (T (a1 ) | · · · | T (an )).)
(a) Exchange the first two entries and fix the 3rd; that is, T ((x1 , x2 , x3 )T ) = (x2 , x1 , x3 )T .
(b) Move the 1st, 2nd and 3rd entries of x to 2nd, 3rd and 1st entries, respectively.
(c) The composition of the actions (a) (first) and (b) above.
4. Express each of the following 3 linear transformations of R2 as a matrix transformation:
(a) reflect each point with respect to the straight line x1 = x2 ;
(b) rotate each point in the counter clockwise direction by π/4;
(c) the composition of the actions (a) (first) and (b) above.
5. Consider a basis of the following 3 vectors of R3
 
 


1
1
1
b1 =  2  , b2 =  2  , b3 =  0  .
3
0
0
(a) Find the B-coordinate of the vector (1, 1, 0)T in the basis above.
(b) Find the change-of-basis matrix PB .
(c) Let E = {e1 , e2 , e3 } denote the standard basis of R3 . Find the change-of-basis matrices
P
P
B ← E and E ← B.
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MATH2076-001, Spring 2022
Sample questions
Note: Given a basis B of Rn , the change-of-basis matrix PB is a n × n matrix such that for every
vector x ∈ Rn and its B-coordinate [x]B ,
x = PB [x]B
with
PB = (b1 | · · · | bn ) .
P
Given two basis B = {b1 , . . . , bn }, C = {c1 , . . . , cn }, the change-of-basis matrix B ← C is the
n × n matrix such that
P
[x]B B ← C [x]C .
There are two methods to compute the change-of-basis matrix as summarized below
P
B ← C = ([c1 ]B | · · · | [cn ]B ) ,
or
P
B ← C = PB−1 PC = (b1 | · · · | bn )−1 (c1 | · · · | cn ).
6. Consider the following three polynomials:
p1 (t) = t2 + t + 1,
p2 (t) = t2 − 2t,
p3 (t) = 3t + 1.
Are they linearly independent?
7. Consider the following two bases of P1 , the vector space of polynomials with degree at most one
and real coefficients:
B = {1, t} ,
C = {1, t + 1} .
(a) Compute directly [t]B , [t]C .
P
(b) Compute the change of basis matrix C ← B.
Note: in this example, for polynomial function p(t) ∈ P1 , [p(t)]B is a 2-dimensional vector
(c1 , c2 )T ∈ R2 such that p(t) = c1 + c2 t. The matrix in part (b) is such that
P
[p(t)]C C ← B [p(t)]B for all p(t) ∈ P1 ,
with
P
C ← B = ([p1 (t)]C | [p2 (t)]C ) if B = {p1 (t), p2 (t)}.
8. Consider the following matrix

1
A= 0
0
2
1
3

3
0 .
4
(a) Compute its determinant: det A.
(b) Compute its characteristic polynomial: pA (λ) = det(A − λI).
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MATH2076-001, Spring 2022
Sample questions
9. Consider the following matrix

1
A= 0
0
2
1
0

0
0 .
3
Does it have a diagonalization A = P DP −1 for some invertible matrix P ? If so compute, if not
explain why.
10. Consider the following symmetric matrix

1
A= 2
0
2
1
0

0
0 .
3
(1)
Does it have an orthogonal diagonalization A = P DP −1 for some orthogonal matrix P ? If so
compute, if not explain why.
Note: A squared matrix P is orthogonal, if and only if its column vectors are orthonormal.
11. Consider a basis of the following 3 vectors of R3


 


1
1
0
b1 =  1  , b2 =  1  , b3 =  1  .
1
0
1
(2)
(a) Provide a family of 3 orthogonal vectors by applying the Gram–Schmidt process to the basis
above.
(b) Using the vectors obtained in the previous part, find the QR factorization of A.
12. Consider the three vectors b1 , b2 , b3 in (2). Find the orthogonal projection of vector b3 onto
the space W = span{b1 , b2 }. What is the approximation error of the projected vector projW b
w.r.t. b?
Note: recall the projection formulas: if a1 , . . . , ap are linearly independent in Rn , and W =
span{a1 , . . . , ap }, then
projW b = A(AT A)−1 AT b.
If in addition the vectors a1 , . . . , ap are orthogonal, then one has in addition
projW x =
x1 · ap
x1 · a1
a1 + · · · +
ap .
a1 · a1
ap · ap
The projection error in this context is kprojW x − xk.
13. Find the matrix A so that xT Ax = Q(x) = x21 +2x22 +3x23 +2x1 x2 −2x1 x3 with x = (x1 , x2 , x3 )T .
14. Consider the following inconsistent linear system:

Ax = b
with
1
A= 1
1
 

1
1
1 ,b =  2 .
0
3
Write down the normal equation, and solve for the least-square solution of the original system.
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MATH2076-001, Spring 2022
Sample questions
15. Consider the quadratic form Q(x) = xT Ax with x = (x1 , x2 , x3 )T and A as in (1). Is Q(x)
positive definite, positive semi-definite, negative definite, negative semi-definite, or indefinite?
What is the smallest possible value Q(x) may achieve for x ∈ R3 ?
16. Find the singular value decomposition of the matrix


3 −3
A =  0 0 .
1 1
4