APPLIED MATHEMATICS 1A (MATH132)
ADDITIONAL PROBLEMS 1
1. Suppose that the augmented matrix for a system of linear equations has been completed
reduced to the following matrix
1 −6 0 0 3 −2
0 0 1 0 4 7
.
0 0 0 1 5 8
0 0 0 0 0 0
Write down the general solution of the linear system. (Ans: x1 = −2 + 6x2 − 3x5 , x2
is free, x3 = 7 − 4x5 , x4 = 8 − 5x5 , x5 is free)
2. Do the three planes 2x2 − 2x3 = 1, x1 + 2x2 − x3 = −3 and x1 − x2 + 2x3 = 3 have at
least one common point of intersection? Explain.
3. Consider the system
x1 + x2 + 2x3 = a
x1 + x3 = b
2x1 + x2 + 3x3 = c
where a, b and c are arbitrary real numbers. For the system to be consistent what
condition must a, b and c satisfy? (Ans: c = a + b)
4. Show that if ad − bc ̸= 0, then the system
ax1 + bx2 = r
cx1 + dx2 = s
has a unique solution.
5. Find all values of k for which the linear system
x1 + 2x2 + x3 = 3
2x1 + 3x2 + 3x3 = 4
3x1 + 3x2 + (k 2 − 3)x3 = k
has (a) no solution, (b) a unique solution and (c) infinitely many solutions.
(Ans: (a) k = −3, (b) k ̸= ±3, (c) k = 3)
6. (a) A system of linear equations with fewer equations than unknowns is often called an
underdetermined system. Suppose that such a system is consistent. Explain why
it must have an infinite number of solutions.
(b) A system of linear equations with more equations than unknowns is often called
an overdetermined system. Show that the following overdetermined system is
consistent.
x1 − 2x2 = 3
3x1 − x2 = 14
x1 − 7x2 = −2
,7. A system of linear equations is called homogeneous if the right hand side of each equa-
tion is zero. Such a system always has at least one solution, i.e. x1 = x2 = · · · = xn = 0.
This is called the trivial solution. All other solutions are called nontrivial solutions.
For what values of k do the following homogeneous systems have nontrivial solutions?
x 1 − x2 + x 3 = 0
(k − 2)x1 + x2 = 0
(a) (b) 3x1 + x2 + 4x3 = 0
x1 + (k − 2)x2 = 0
4x1 + 12x2 + kx3 = 0
(Ans: (a) k = 1 or k = 3, (b) k = 8)
8. (a) Solve the system
x1 + 2x2 + x3 = 3
−3x1 − x2 + 2x3 = 1
5x2 + 3x3 = 2
using the Gauss–Jordan elimination method (i.e. transforming the augmented ma-
trix to reduced row echelon form). (Ans: (x1 , x2 , x3 ) = (3, −2, 4))
−2
(b) Use your answer in part (a) to write the vector 1 as a linear combination of
−5
1 1 3
the vectors −3 , 2 and 1 .
0 3 2
, APPLIED MATHEMATICS 1A (MATH132)
ADDITIONAL PROBLEMS 2
1 2 4 3
1. Let v1 = 0 , v2 = 1 , v3 = 2 and w = 1 .
−1 3 6 2
(a) How many vectors are in {v1 , v2 , v3 }? Is w in {v1 , v2 , v3 }?
(b) How many vectors are in Span {v1 , v2 , v3 }?
(c) Is w in Span {v1 , v2 , v3 }? Why? (Ans: Yes, since w can be expressed as a linear
combination of v1 , v2 and v3 )
1 3 −1 2
2. Consider matrix A = 2 5 −1 and vector b = 1 .
2 8 −2 3
(a) Show that b is a linear combination of the column vectors a1 , a2 and a3 of A.
(b) List three other vectors in Span {a1 , a2 , a3 }.
2 4 x
3. Let v1 = −1 and v2 = 1 be vectors in R . Furthermore let v =
3 y be a
4 6 z
vector in R such that v ∈ Span {v1 , v2 }. Find the equation that describes Span {v1 , v2 }.
3
(Ans: 5x − 2y − 3z = 0)
4 1 2
4. Determine the value of c so that c is in the plane spanned by 1 and −1 .
−3 −2 5
(Ans: 7/3)
1 −2 3 8
5. Let A = 2 −3 2 and b = 7 . Solve the equation Ax = b. Use your answer
−1 1 2 4
to solve the nonlinear system
√
xy − 2 y + 3yz = 8
√
2xy − 3 y + 2yz = 7
√
−xy + y + 2yz = 4
for x, y and z. (Ans: x = 5/9, y = 9, z = 1/3)
2 1 1
6. Let u = 1 , v = 3 and w = −7 . Show that 2u − 3v − w = 0. Hence
1 2 −4
(without using row operations) find x1 and x2 that satisfy the equation
1 2 [ ] 1
−7 1 x1 = 3 .
x2
−4 1 2
(Ans: x1 = −1/3, x2 = 2/3)
ADDITIONAL PROBLEMS 1
1. Suppose that the augmented matrix for a system of linear equations has been completed
reduced to the following matrix
1 −6 0 0 3 −2
0 0 1 0 4 7
.
0 0 0 1 5 8
0 0 0 0 0 0
Write down the general solution of the linear system. (Ans: x1 = −2 + 6x2 − 3x5 , x2
is free, x3 = 7 − 4x5 , x4 = 8 − 5x5 , x5 is free)
2. Do the three planes 2x2 − 2x3 = 1, x1 + 2x2 − x3 = −3 and x1 − x2 + 2x3 = 3 have at
least one common point of intersection? Explain.
3. Consider the system
x1 + x2 + 2x3 = a
x1 + x3 = b
2x1 + x2 + 3x3 = c
where a, b and c are arbitrary real numbers. For the system to be consistent what
condition must a, b and c satisfy? (Ans: c = a + b)
4. Show that if ad − bc ̸= 0, then the system
ax1 + bx2 = r
cx1 + dx2 = s
has a unique solution.
5. Find all values of k for which the linear system
x1 + 2x2 + x3 = 3
2x1 + 3x2 + 3x3 = 4
3x1 + 3x2 + (k 2 − 3)x3 = k
has (a) no solution, (b) a unique solution and (c) infinitely many solutions.
(Ans: (a) k = −3, (b) k ̸= ±3, (c) k = 3)
6. (a) A system of linear equations with fewer equations than unknowns is often called an
underdetermined system. Suppose that such a system is consistent. Explain why
it must have an infinite number of solutions.
(b) A system of linear equations with more equations than unknowns is often called
an overdetermined system. Show that the following overdetermined system is
consistent.
x1 − 2x2 = 3
3x1 − x2 = 14
x1 − 7x2 = −2
,7. A system of linear equations is called homogeneous if the right hand side of each equa-
tion is zero. Such a system always has at least one solution, i.e. x1 = x2 = · · · = xn = 0.
This is called the trivial solution. All other solutions are called nontrivial solutions.
For what values of k do the following homogeneous systems have nontrivial solutions?
x 1 − x2 + x 3 = 0
(k − 2)x1 + x2 = 0
(a) (b) 3x1 + x2 + 4x3 = 0
x1 + (k − 2)x2 = 0
4x1 + 12x2 + kx3 = 0
(Ans: (a) k = 1 or k = 3, (b) k = 8)
8. (a) Solve the system
x1 + 2x2 + x3 = 3
−3x1 − x2 + 2x3 = 1
5x2 + 3x3 = 2
using the Gauss–Jordan elimination method (i.e. transforming the augmented ma-
trix to reduced row echelon form). (Ans: (x1 , x2 , x3 ) = (3, −2, 4))
−2
(b) Use your answer in part (a) to write the vector 1 as a linear combination of
−5
1 1 3
the vectors −3 , 2 and 1 .
0 3 2
, APPLIED MATHEMATICS 1A (MATH132)
ADDITIONAL PROBLEMS 2
1 2 4 3
1. Let v1 = 0 , v2 = 1 , v3 = 2 and w = 1 .
−1 3 6 2
(a) How many vectors are in {v1 , v2 , v3 }? Is w in {v1 , v2 , v3 }?
(b) How many vectors are in Span {v1 , v2 , v3 }?
(c) Is w in Span {v1 , v2 , v3 }? Why? (Ans: Yes, since w can be expressed as a linear
combination of v1 , v2 and v3 )
1 3 −1 2
2. Consider matrix A = 2 5 −1 and vector b = 1 .
2 8 −2 3
(a) Show that b is a linear combination of the column vectors a1 , a2 and a3 of A.
(b) List three other vectors in Span {a1 , a2 , a3 }.
2 4 x
3. Let v1 = −1 and v2 = 1 be vectors in R . Furthermore let v =
3 y be a
4 6 z
vector in R such that v ∈ Span {v1 , v2 }. Find the equation that describes Span {v1 , v2 }.
3
(Ans: 5x − 2y − 3z = 0)
4 1 2
4. Determine the value of c so that c is in the plane spanned by 1 and −1 .
−3 −2 5
(Ans: 7/3)
1 −2 3 8
5. Let A = 2 −3 2 and b = 7 . Solve the equation Ax = b. Use your answer
−1 1 2 4
to solve the nonlinear system
√
xy − 2 y + 3yz = 8
√
2xy − 3 y + 2yz = 7
√
−xy + y + 2yz = 4
for x, y and z. (Ans: x = 5/9, y = 9, z = 1/3)
2 1 1
6. Let u = 1 , v = 3 and w = −7 . Show that 2u − 3v − w = 0. Hence
1 2 −4
(without using row operations) find x1 and x2 that satisfy the equation
1 2 [ ] 1
−7 1 x1 = 3 .
x2
−4 1 2
(Ans: x1 = −1/3, x2 = 2/3)
