SCHOOL OF EDUCATION AND SOCIAL SCIENCES
DEPARTMENT OF EDUCATION
UNIT TITLE: LINEAR ALGEBRA 1
UNIT CODE: MATH 207
Author: DAVID MURIUKI
,The only limit of our realization of tomorrow will be our doubts of today by David
Muriuki
Course Objectives
The purpose of the course is to introduce learners to the concept of vectors,
vector spaces, matrices, solution of linear systems and linear transformations.
Expected Learning Outcomes
By the end of the course, Learners should be able to:
i. Solve systems of linear equations in several variables.
ii. perform elementary operations on vectors and determine if given vectors are
linearly independent or dependent.
iii. Determine if a given transformation is linear or not.
iv. Determine if given sets are subspaces of vector space or not.
iv. Find the matrix representation of a linear transformation relative to a given
basis.
Course Content
3
Vectors and scalars: algebraic and geometric properties in Â
magnitude and directions and applications and applications such as force, dis-
placement, velocity and rotation;
Operations: On vectors such as addition, scalar multiplication, dot and cross
products, and linear combination, vector proof of theorems in geometry.
Extension to Ân : scalar products, formulae for length and angle, Schwartz and
triangular inequalities, and
planes and lines in Â3 ;
Vector space over Ân ; Subspace, spanning sets, linear independence, bases and
dimension, direct sums and intersection of subspace, linear mapping and their
matrices with respect to the standard basis, range and nullspace, nullity and
echelon form, rotations and reflections in Â2 and Â3 , and application to linear
equations, matrix; Multiplication, inverse mappings and their matrices
Learning and Teaching Methodology
,The only limit of our realization of tomorrow will be our doubts of today by David
Muriuki
Lectures, tutorials, Internet, resource persons, group discussions.
Instructional Materials/ Equipment
Power point, white boards, hand-outs, transparencies, charts/maps, DVDs.
1 LESSON ONE: LINEAR SYSTEMS
1.1 Introduction to Systems of Linear Equations
Recall that in two dimensions a line in a rectangular xy-coordinate system can
be represented by an equation of the form
ax + by = c (a, b not both 0)
and in three dimensions a plane in a rectangular xyz−coordinate system can be
represented by an equation of the form
ax + by + cz = d (a, b, c not all 0)
These are examples of “linear equations,” the first being a linear equation in the
variablesx and y and the second a linear equation in the variablesx, y, and z.
More generally, we define a linear equation in the n variables x1 , x2 , ..., xn to be
one that can be expressed in the form
a1 x1 + a2 x2 + · · · + an xn = b...................................(1)
where a1 , a2 , ..., an and b are constants, and the a’s are not all zero. In the special
cases where n = 2 orn = 3, we will often use variables without subscripts and
, The only limit of our realization of tomorrow will be our doubts of today by David
Muriuki
write linear equations as
a1 x + a2 y = b (a1 , a2 not both 0).......................(2)
a1 x + a2 y + a3 z = b (a1 , a2 , a3 not all 0)..................(3)
In the special case whereb = 0, Equation (1) has the form
a1 x1 + a2 x2 + · · · + an xn = 0.............................(4)
which is called a homogeneous linear equation in the variables x1 , x2 , ..., xn .
EXAMPLE 1 Linear Equations
A linear equation does not involve any products or roots of variables. All
variables occur only to the first power and do not appear, for example, as ar-
guments of trigonometric, logarithmic, or exponential functions. The following
are linear equations:
x + 3y = 7
1
x−y + 3z = −1
2
x1 −2x2 −3x3 + x4 = 0
−1x1 + x2 + ··· + xn = 1
The following are not linear equations:
x + 3y 2 = 4
3x + 2y−xy = 5
sinx + y = 0
√
x1 + 2x2 + x3 = 1
DEPARTMENT OF EDUCATION
UNIT TITLE: LINEAR ALGEBRA 1
UNIT CODE: MATH 207
Author: DAVID MURIUKI
,The only limit of our realization of tomorrow will be our doubts of today by David
Muriuki
Course Objectives
The purpose of the course is to introduce learners to the concept of vectors,
vector spaces, matrices, solution of linear systems and linear transformations.
Expected Learning Outcomes
By the end of the course, Learners should be able to:
i. Solve systems of linear equations in several variables.
ii. perform elementary operations on vectors and determine if given vectors are
linearly independent or dependent.
iii. Determine if a given transformation is linear or not.
iv. Determine if given sets are subspaces of vector space or not.
iv. Find the matrix representation of a linear transformation relative to a given
basis.
Course Content
3
Vectors and scalars: algebraic and geometric properties in Â
magnitude and directions and applications and applications such as force, dis-
placement, velocity and rotation;
Operations: On vectors such as addition, scalar multiplication, dot and cross
products, and linear combination, vector proof of theorems in geometry.
Extension to Ân : scalar products, formulae for length and angle, Schwartz and
triangular inequalities, and
planes and lines in Â3 ;
Vector space over Ân ; Subspace, spanning sets, linear independence, bases and
dimension, direct sums and intersection of subspace, linear mapping and their
matrices with respect to the standard basis, range and nullspace, nullity and
echelon form, rotations and reflections in Â2 and Â3 , and application to linear
equations, matrix; Multiplication, inverse mappings and their matrices
Learning and Teaching Methodology
,The only limit of our realization of tomorrow will be our doubts of today by David
Muriuki
Lectures, tutorials, Internet, resource persons, group discussions.
Instructional Materials/ Equipment
Power point, white boards, hand-outs, transparencies, charts/maps, DVDs.
1 LESSON ONE: LINEAR SYSTEMS
1.1 Introduction to Systems of Linear Equations
Recall that in two dimensions a line in a rectangular xy-coordinate system can
be represented by an equation of the form
ax + by = c (a, b not both 0)
and in three dimensions a plane in a rectangular xyz−coordinate system can be
represented by an equation of the form
ax + by + cz = d (a, b, c not all 0)
These are examples of “linear equations,” the first being a linear equation in the
variablesx and y and the second a linear equation in the variablesx, y, and z.
More generally, we define a linear equation in the n variables x1 , x2 , ..., xn to be
one that can be expressed in the form
a1 x1 + a2 x2 + · · · + an xn = b...................................(1)
where a1 , a2 , ..., an and b are constants, and the a’s are not all zero. In the special
cases where n = 2 orn = 3, we will often use variables without subscripts and
, The only limit of our realization of tomorrow will be our doubts of today by David
Muriuki
write linear equations as
a1 x + a2 y = b (a1 , a2 not both 0).......................(2)
a1 x + a2 y + a3 z = b (a1 , a2 , a3 not all 0)..................(3)
In the special case whereb = 0, Equation (1) has the form
a1 x1 + a2 x2 + · · · + an xn = 0.............................(4)
which is called a homogeneous linear equation in the variables x1 , x2 , ..., xn .
EXAMPLE 1 Linear Equations
A linear equation does not involve any products or roots of variables. All
variables occur only to the first power and do not appear, for example, as ar-
guments of trigonometric, logarithmic, or exponential functions. The following
are linear equations:
x + 3y = 7
1
x−y + 3z = −1
2
x1 −2x2 −3x3 + x4 = 0
−1x1 + x2 + ··· + xn = 1
The following are not linear equations:
x + 3y 2 = 4
3x + 2y−xy = 5
sinx + y = 0
√
x1 + 2x2 + x3 = 1
