MA222
Week 1 Complete Mastery Guide
Lectures 1 & 2: Vector Spaces, Linear Independence, Bases,
Wronskian
! Vector spaces and subspaces (revision + MA222 perspective)
! Linear independence — columns vs rows method
! Bases, dimension, coordinate vectors
! Function spaces and the Wronskian test
! Transcript patches: inspection method, Wronskian counterexample
Sources: Lecture 1–2 slides, Lecture Notes Ch. 1–2, Transcripts 230120 &
270120,
Exercises 1 solutions, Past papers 2017/2024/2025/IRDAP
,Contents
A. Core Theory (with Integrated Worked Examples) 3
1. Vector Spaces: The Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3. Linear Span, Bases, Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4. The Wronskian Test for Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 5
B. Rigorous Definitions 7
C. Key Theorems 7
D. Formula Sheet 8
E. Exam Questions 9
F. Worked Solutions 10
Solution 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Solution 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Solution 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
G. Common Mistakes 10
H. One-Page Cheat Sheet 11
Additional Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2
,A. Core Theory (with Integrated Worked Examples)
1. Vector Spaces: The Setting
Everything in MA222 happens inside a vector space. You know these from MA100,
but Dr Ward wants you to think about them more fluidly now.
Definition: Vector Space
A vector space V over R (or C) is a set with two operations — vector addi-
tion and scalar multiplication — satisfying: closure under both operations, ex-
istence of zero vector, existence of additive inverses, and the usual associativ-
ity/commutativity/distributivity axioms.
Key examples for this course:
• Rn : column vectors with real entries (the workhorse)
• C[0, 1]: continuous functions on [0, 1] (used for inner products, Gram-Schmidt)
• Pn : polynomials of degree → n (used for Wronskian, function space questions)
• Cn : column vectors with complex entries (Weeks 4–5 onward)
[Lecture 1, pp. 2–5; Lecture Notes Section 1.1]
Definition: Subspace
A non-empty subset U of a vector space V is a subspace if it is closed under
vector addition and scalar multiplication. Equivalently: (1) 0 ∈ U , (2) u1 , u2 ∈ U ⇒
u1 + u2 ∈ U , (3) α ∈ R, u ∈ U ⇒ αu ∈ U .
Worked Example: Subspace Verification
Show U = {(x, y, z)t ∈ R3 : 2x − y + z = 0} is a subspace of R3 .
Step 1 (Non-empty): 0 = (0, 0, 0)t : 2(0) − 0 + 0 = 0. !
Step 2 (CUVA): Take u = (x1 , y1 , z1 )t , v = (x2 , y2 , z2 )t ∈ U . Then 2x1 − y1 + z1 = 0 and
2x2 −y2 +z2 = 0. Their sum: 2(x1 +x2 )−(y1 +y2 )+(z1 +z2 ) = (2x1 −y1 +z1 )+(2x2 −y2 +z2 ) =
0. !
Step 3 (CUSM): αu: 2(αx1 ) − αy1 + αz1 = α(2x1 − y1 + z1 ) = 0. !
All three pass ⇒ U is a subspace. "
NOT a Subspace: The “+1” Trap
{(x, y) ∈ R2 : y = 2x + 1} is NOT a subspace. Quick check: (0, 0) doesn’t satisfy
0 = 2(0) + 1, so 0 ∈/ U . Any set defined by a non-homogeneous equation
fails.
3
, 2. Linear Independence
Definition: Linear Independence
Vectors v1 , . . . , vk are linearly independent (LI) if α1 v1 + · · · + αk vk = 0 implies
α1 = · · · = αk = 0. Otherwise, they are linearly dependent (LD).
METHOD: Row Reduction for LI (Dr Ward’s Preferred Approach)
In MA100, you put vectors as columns and check column rank. In MA222, Dr
Ward prefers putting vectors as rows:
Step 1: Write vectors as rows of a matrix.
Step 2: Row reduce to echelon form.
Step 3: Row of zeros ⇒ LD. No row of zeros ⇒ LI.
Why this is better (Lecture 1 transcript): “We’re not little robots now. We’re not
doing MA100 for the first time.” The non-zero rows of the echelon form give you
a nicer basis for the same span — they have more zeros, making them easier
to work with in later calculations (projections, etc.).
Worked Example: LI in R3
Test {(1, 3, 1)t , (0, 2, 1)t , (2, 0, −1)t } for LI.
Put as rows:
) ) )
1 3 1 1 3 1 1 3 1
R3 →2R1 R3 +3R2
0 2 1 −−−−−→ 0 2 1 −−−−−→ 0 2 1
2 0 −1 0 −6 −3 0 0 0
Row of zeros ⇒ LD. Two non-zero rows ⇒ span is 2-dimensional. Basis:
{(1, 3, 1)t , (0, 2, 1)t }.
EXAM SPEED: Inspection First (Dr Ward, Lectures 1–2)
Before row-reducing, look at the vectors:
— Two vectors? Just check if one is a scalar multiple of the other.
— n vectors in Rn ? Compute the determinant. Non-zero ⇔ LI.
— Spot a vector that’s a sum/difference of others? They’re LD.
“If you can see a quicker way, we can do the quicker way.” Row operations are
the fallback, not the default.
4
Week 1 Complete Mastery Guide
Lectures 1 & 2: Vector Spaces, Linear Independence, Bases,
Wronskian
! Vector spaces and subspaces (revision + MA222 perspective)
! Linear independence — columns vs rows method
! Bases, dimension, coordinate vectors
! Function spaces and the Wronskian test
! Transcript patches: inspection method, Wronskian counterexample
Sources: Lecture 1–2 slides, Lecture Notes Ch. 1–2, Transcripts 230120 &
270120,
Exercises 1 solutions, Past papers 2017/2024/2025/IRDAP
,Contents
A. Core Theory (with Integrated Worked Examples) 3
1. Vector Spaces: The Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3. Linear Span, Bases, Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4. The Wronskian Test for Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 5
B. Rigorous Definitions 7
C. Key Theorems 7
D. Formula Sheet 8
E. Exam Questions 9
F. Worked Solutions 10
Solution 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Solution 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Solution 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
G. Common Mistakes 10
H. One-Page Cheat Sheet 11
Additional Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2
,A. Core Theory (with Integrated Worked Examples)
1. Vector Spaces: The Setting
Everything in MA222 happens inside a vector space. You know these from MA100,
but Dr Ward wants you to think about them more fluidly now.
Definition: Vector Space
A vector space V over R (or C) is a set with two operations — vector addi-
tion and scalar multiplication — satisfying: closure under both operations, ex-
istence of zero vector, existence of additive inverses, and the usual associativ-
ity/commutativity/distributivity axioms.
Key examples for this course:
• Rn : column vectors with real entries (the workhorse)
• C[0, 1]: continuous functions on [0, 1] (used for inner products, Gram-Schmidt)
• Pn : polynomials of degree → n (used for Wronskian, function space questions)
• Cn : column vectors with complex entries (Weeks 4–5 onward)
[Lecture 1, pp. 2–5; Lecture Notes Section 1.1]
Definition: Subspace
A non-empty subset U of a vector space V is a subspace if it is closed under
vector addition and scalar multiplication. Equivalently: (1) 0 ∈ U , (2) u1 , u2 ∈ U ⇒
u1 + u2 ∈ U , (3) α ∈ R, u ∈ U ⇒ αu ∈ U .
Worked Example: Subspace Verification
Show U = {(x, y, z)t ∈ R3 : 2x − y + z = 0} is a subspace of R3 .
Step 1 (Non-empty): 0 = (0, 0, 0)t : 2(0) − 0 + 0 = 0. !
Step 2 (CUVA): Take u = (x1 , y1 , z1 )t , v = (x2 , y2 , z2 )t ∈ U . Then 2x1 − y1 + z1 = 0 and
2x2 −y2 +z2 = 0. Their sum: 2(x1 +x2 )−(y1 +y2 )+(z1 +z2 ) = (2x1 −y1 +z1 )+(2x2 −y2 +z2 ) =
0. !
Step 3 (CUSM): αu: 2(αx1 ) − αy1 + αz1 = α(2x1 − y1 + z1 ) = 0. !
All three pass ⇒ U is a subspace. "
NOT a Subspace: The “+1” Trap
{(x, y) ∈ R2 : y = 2x + 1} is NOT a subspace. Quick check: (0, 0) doesn’t satisfy
0 = 2(0) + 1, so 0 ∈/ U . Any set defined by a non-homogeneous equation
fails.
3
, 2. Linear Independence
Definition: Linear Independence
Vectors v1 , . . . , vk are linearly independent (LI) if α1 v1 + · · · + αk vk = 0 implies
α1 = · · · = αk = 0. Otherwise, they are linearly dependent (LD).
METHOD: Row Reduction for LI (Dr Ward’s Preferred Approach)
In MA100, you put vectors as columns and check column rank. In MA222, Dr
Ward prefers putting vectors as rows:
Step 1: Write vectors as rows of a matrix.
Step 2: Row reduce to echelon form.
Step 3: Row of zeros ⇒ LD. No row of zeros ⇒ LI.
Why this is better (Lecture 1 transcript): “We’re not little robots now. We’re not
doing MA100 for the first time.” The non-zero rows of the echelon form give you
a nicer basis for the same span — they have more zeros, making them easier
to work with in later calculations (projections, etc.).
Worked Example: LI in R3
Test {(1, 3, 1)t , (0, 2, 1)t , (2, 0, −1)t } for LI.
Put as rows:
) ) )
1 3 1 1 3 1 1 3 1
R3 →2R1 R3 +3R2
0 2 1 −−−−−→ 0 2 1 −−−−−→ 0 2 1
2 0 −1 0 −6 −3 0 0 0
Row of zeros ⇒ LD. Two non-zero rows ⇒ span is 2-dimensional. Basis:
{(1, 3, 1)t , (0, 2, 1)t }.
EXAM SPEED: Inspection First (Dr Ward, Lectures 1–2)
Before row-reducing, look at the vectors:
— Two vectors? Just check if one is a scalar multiple of the other.
— n vectors in Rn ? Compute the determinant. Non-zero ⇔ LI.
— Spot a vector that’s a sum/difference of others? They’re LD.
“If you can see a quicker way, we can do the quicker way.” Row operations are
the fallback, not the default.
4
