MATHS 244 LINEAR ALGEBRA
Revision
Systems of Linear Equations & Matrices
A system of linear equations is a collection of equations that share the same set of
unknowns.
Every system can be represented compactly in matrix form:
𝐴𝐱 = 𝐛
where:
• 𝐴= coefficient matrix
• 𝐱= column vector of variables
• 𝐛= column vector of constants. This form unifies all equations into one clean
algebraic statement.
For example, the system
𝑥 + 2𝑦 − 𝑧 = 1
{
2𝑥 + 3𝑦 + 𝑧 = 4
can be written as:
𝑥
1 2 −1 𝑦 1
[ ] ] = [ ] 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑖𝑠 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚 𝐴𝒙 = 𝒃
[
1 3 1 4
𝑧
This compact form is essential for solving systems using matrix methods.
Solving Systems: Gaussian Elimination
Gaussian elimination (or row reduction) is a method to solve systems by transforming
the augmented matrix [ 𝐴 ∣ 𝐛 ] into Reduced Row Echelon Form through elementary row
operations:
1. Swap two rows
2. Multiply a row by a nonzero scalar
3. Add or subtract a multiple of one row from another
, The goal:
Make each pivot (leading nonzero entry in a row) equal to 1 and make everything above
and below each pivot 0.
Example:
1 2 −1 | 1 1 0 −5 | −2
[ ]→[ ]
2 3 1 |4 0 1 2| 3
This final form directly gives the solutions for 𝑥, 𝑦, 𝑧.
Pivots, Leading Variables, and Free Variables
• A pivot position is the location of the leading 1 in each row after row reduction.
• The variables corresponding to pivot columns are leading variables.
• The remaining variables (those without pivots) are free variables.
Free variables can take any value → if at least one exists, the system has infinitely many
solutions.
Types of Solutions
Type Condition Description
One pivot per column, no Exactly one intersection
Unique
contradictions point
Infinitely many At least one free variable Flat (line/plane) of solutions
A row like [0 0 0 | c]
Inconsistent No solution
where c is a constant
Homogeneous Systems
A homogeneous system has 𝐛 = 𝟎:
𝐴𝐱 = 𝟎
These systems are always consistent because 𝐱 = 𝟎 is a solution (called the trivial
solution).
If there are free variables, there are also non-trivial solutions (infinitely many).
• If all variables are leading → only trivial solution.
, • If free variables exist → infinitely many non-trivial solutions.
General Solution Form
• Two independent equations in ℝ2 intersect at one point → unique solution.
• Two parallel equations → no solution.
• Same equation twice → infinite solutions.
If the system has infinitely many solutions, they can be expressed as:
𝐱 = 𝐱 𝐩 + 𝑡1 𝐯𝟏 + 𝑡2 𝐯𝟐 + ⋯
where
• 𝐱 𝐩= a particular solution,
• 𝐯𝐢= vectors from the null space of 𝐴,
• 𝑡𝑖 = free parameters.
, Vectors and Vector Spaces
What is a Vector?
A vector is a mathematical object that has both magnitude and direction.
In linear algebra, we generalize this idea:
• A vector can be an ordered list of numbers (e.g. 𝐯 = [2,–1,4]),
• or a function, polynomial, or any object that behaves like one under addition and
scalar multiplication.
A vector space is the“world” these vectors live in - a set of vectors that can be added
together and scaled by numbers (scalars) without leaving the space.
Vector Space Axioms
A set 𝑉 is a vector space over a field 𝐹 (usually ℝ or ℂ) if it satisfies these 10 axioms:
1. Closure under addition: 𝐮, 𝐯 ∈ 𝑉 ⇒ 𝐮 + 𝐯 ∈ 𝑉
2. Commutativity: 𝐮+𝐯 =𝐯+𝐮
3. Associativity: (𝐮 + 𝐯) + 𝐰 = 𝐮 + (𝐯 + 𝐰)
4. Additive identity: There exists 𝟎 such that 𝐯 + 𝟎 = 𝐯
5. Additive inverse: For each 𝐯 there exists −𝐯 such that 𝐯 + (−𝐯) = 0
6. Closure under scalar multiplication: 𝑐𝐯 ∈ 𝑉 for all 𝑐 ∈ 𝐹
7. Distributivity (I): 𝑐(𝐮 + 𝐯) = 𝑐𝐮 + 𝑐𝐯
8. Distributivity (II): (𝑐 + 𝑑)𝐯 = 𝑐𝐯 + 𝑑𝐯
9. Associativity of scalar multiplication: 𝑐(𝑑𝐯) = (𝑐𝑑)𝐯
10. Identity element of scalar multiplication: 1𝐯 = 𝐯
A field (denoted 𝐹) is the set of numbers you use for scalars in a vector space — the
values that multiply your vectors with.
Examples include real, rational and complex numbers (ℝ, ℚ, ℂ)
Subspaces
A subspace 𝑊of 𝑉 is a subset that is itself a vector space under the same operations.
To check if 𝑊 is a subspace:
Revision
Systems of Linear Equations & Matrices
A system of linear equations is a collection of equations that share the same set of
unknowns.
Every system can be represented compactly in matrix form:
𝐴𝐱 = 𝐛
where:
• 𝐴= coefficient matrix
• 𝐱= column vector of variables
• 𝐛= column vector of constants. This form unifies all equations into one clean
algebraic statement.
For example, the system
𝑥 + 2𝑦 − 𝑧 = 1
{
2𝑥 + 3𝑦 + 𝑧 = 4
can be written as:
𝑥
1 2 −1 𝑦 1
[ ] ] = [ ] 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑖𝑠 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚 𝐴𝒙 = 𝒃
[
1 3 1 4
𝑧
This compact form is essential for solving systems using matrix methods.
Solving Systems: Gaussian Elimination
Gaussian elimination (or row reduction) is a method to solve systems by transforming
the augmented matrix [ 𝐴 ∣ 𝐛 ] into Reduced Row Echelon Form through elementary row
operations:
1. Swap two rows
2. Multiply a row by a nonzero scalar
3. Add or subtract a multiple of one row from another
, The goal:
Make each pivot (leading nonzero entry in a row) equal to 1 and make everything above
and below each pivot 0.
Example:
1 2 −1 | 1 1 0 −5 | −2
[ ]→[ ]
2 3 1 |4 0 1 2| 3
This final form directly gives the solutions for 𝑥, 𝑦, 𝑧.
Pivots, Leading Variables, and Free Variables
• A pivot position is the location of the leading 1 in each row after row reduction.
• The variables corresponding to pivot columns are leading variables.
• The remaining variables (those without pivots) are free variables.
Free variables can take any value → if at least one exists, the system has infinitely many
solutions.
Types of Solutions
Type Condition Description
One pivot per column, no Exactly one intersection
Unique
contradictions point
Infinitely many At least one free variable Flat (line/plane) of solutions
A row like [0 0 0 | c]
Inconsistent No solution
where c is a constant
Homogeneous Systems
A homogeneous system has 𝐛 = 𝟎:
𝐴𝐱 = 𝟎
These systems are always consistent because 𝐱 = 𝟎 is a solution (called the trivial
solution).
If there are free variables, there are also non-trivial solutions (infinitely many).
• If all variables are leading → only trivial solution.
, • If free variables exist → infinitely many non-trivial solutions.
General Solution Form
• Two independent equations in ℝ2 intersect at one point → unique solution.
• Two parallel equations → no solution.
• Same equation twice → infinite solutions.
If the system has infinitely many solutions, they can be expressed as:
𝐱 = 𝐱 𝐩 + 𝑡1 𝐯𝟏 + 𝑡2 𝐯𝟐 + ⋯
where
• 𝐱 𝐩= a particular solution,
• 𝐯𝐢= vectors from the null space of 𝐴,
• 𝑡𝑖 = free parameters.
, Vectors and Vector Spaces
What is a Vector?
A vector is a mathematical object that has both magnitude and direction.
In linear algebra, we generalize this idea:
• A vector can be an ordered list of numbers (e.g. 𝐯 = [2,–1,4]),
• or a function, polynomial, or any object that behaves like one under addition and
scalar multiplication.
A vector space is the“world” these vectors live in - a set of vectors that can be added
together and scaled by numbers (scalars) without leaving the space.
Vector Space Axioms
A set 𝑉 is a vector space over a field 𝐹 (usually ℝ or ℂ) if it satisfies these 10 axioms:
1. Closure under addition: 𝐮, 𝐯 ∈ 𝑉 ⇒ 𝐮 + 𝐯 ∈ 𝑉
2. Commutativity: 𝐮+𝐯 =𝐯+𝐮
3. Associativity: (𝐮 + 𝐯) + 𝐰 = 𝐮 + (𝐯 + 𝐰)
4. Additive identity: There exists 𝟎 such that 𝐯 + 𝟎 = 𝐯
5. Additive inverse: For each 𝐯 there exists −𝐯 such that 𝐯 + (−𝐯) = 0
6. Closure under scalar multiplication: 𝑐𝐯 ∈ 𝑉 for all 𝑐 ∈ 𝐹
7. Distributivity (I): 𝑐(𝐮 + 𝐯) = 𝑐𝐮 + 𝑐𝐯
8. Distributivity (II): (𝑐 + 𝑑)𝐯 = 𝑐𝐯 + 𝑑𝐯
9. Associativity of scalar multiplication: 𝑐(𝑑𝐯) = (𝑐𝑑)𝐯
10. Identity element of scalar multiplication: 1𝐯 = 𝐯
A field (denoted 𝐹) is the set of numbers you use for scalars in a vector space — the
values that multiply your vectors with.
Examples include real, rational and complex numbers (ℝ, ℚ, ℂ)
Subspaces
A subspace 𝑊of 𝑉 is a subset that is itself a vector space under the same operations.
To check if 𝑊 is a subspace:
