Algebra practice exercises and hints to tutorials
These practice exercises should not substitute tutorials, NO!
You must do all tutorial questions in Ridley. Besides tutorials, you should attempt
these exercises. If you are stuck, feel free to consult a lecturer (see the consultation
timetable).
First, I will give hints to tutorials:
Hints to Tutorials
ˆ Question 3, page 103:
Create a second equation by conjugating the first equation α + αβ + β = 0 (1).
After conjugating, you obtain the second equation, call it equation (2). Now treat
β and β as unknowns (like x and y) in the system of equations :(1) and (2). Then
you can use any method to solve for the variables β and β. If you use the inverse
method or Cramer’s rule, the condition |α| 6= 1 ensures that the coefficient matrix
is invertible (has a non-zero determinant). You are not restricted to these methods,
you can use any other relevant method.
ˆ Question 6, page
103:
P
P2N
(n−m)j
=0
(i) The series j=1 (n−m)j on the left-hand side of the equation 2N
j=1 n−m
is a geometric series with common ration and the number of terms is 2N , first
term is n−m . The formula for a finite geometric series is:
(first term) 1 − (common ratio)number of terms
.
1 − common ratio
Also note that (n−m)2N = (2N )n−m
(ii) For orthonormal basis; look at two cases: n = m (doting a vector by itself)
and n 6= m (two different vectors)
ˆ Question 7, page 103:
Given that e1 , e2 , e3 , . . . , en be an orthonormal basis of Cn . Then every vector a
can be written as a linear combination of these vectors, i.e.
a = λ1 e1 + λ2 e2 + λ3 e3 + · · · λn en
1
According to the hint,
a · a = (λ1 e1 + λ2 e2 + λ3 e3 + · · · + λn en ) · a
= (λ1 e1 ) · a + (λ2 e2 ) · a + (λ3 e3 ) · a + · · · + (λn en ) · a
= λ1 (e1 · a) + λ2 (e2 · a) + λ3 (e3 · a) + · · · + λn (en · a)
(0.1)
By the theorem on page 102, we know how to find the coefficients λi ’s. The formula
is given by:
λi = a · e i
Now, substituting this value of λi (for each i = 1, 2, 3, . . . , n) in equation (0.1)
yields:
a · a = (a · e1 )(e1 · a) + (a · e2 )(e2 · a) + (a · e3 )(e3 · a) + · · · + (a · en )(en · a)
From here, you may use the property that for vectors u and v, we have
v·u=u·v
and for every complex number z,
zz = |z|2 .
Practice exercises
1. For a, b and c in Cn , prove that the following properties hold:
(i) a · (b + c) = a · b + a · c
(ii) (a + b) · c = a · c + b · c
Hint: Start with setting a = (a1 , a2 , a3 , . . . , an ), b = (b1 , b2 , b3 , . . . , bn ) and
c = (c1 , c2 , c3 , . . . , cn ). Then work out the LHS (left-hand side) and thereafter,
the RHS (right-hand side). Ensure that you get the same expressions. For
example, the brackets indicate what operation should come FIRST.
2. Suppose that x, y ∈ R. Is it possible for the vectors (1, x + 1, y) and (2 − y, x, i) to
be orthogonal? Justify your answer.
3. Suppose that x, y ∈ R and that the vectors (1, x + 1, y) and (2 − y, x, i) are unit
vectors. Show that x and y satisfy the equation 2x + 4y = 3.
4. Use a = (1, i + 1, 0, i) and b = (2i, 6, i, 4) to verify that the dot product can be
represented as a matrix product:
T
a · b = ab


 
2
i−1
1
 0 

 
5. Use A = 
 3i  and B = 3i to verify that the dot product can be represented
4i
1 + 2i
as a matrix product:
A · B = AT B.
2