LINA01 LINEAR ALGEBRA VERIFIED EXAM SOLUTIONS - COMPREHENSIVE
QUESTIONS AND ANSWERS - CURRENT VERSION (2026/2027)
1. Q: What is a vector? ANSWER A vector is a mathematical object with
both magnitude and direction, typically represented as an ordered list of
numbers (components) in n-dimensional space.
2. Q: What is the difference between a row vector and a column vector?
ANSWER A row vector is written horizontally as [a₁, a₂, ..., aₙ], while a column
vector is written vertically. They are transposes of each other.
3. Q: How do you add two vectors? ANSWER Add corresponding
components: if u = (u₁, u₂, ..., uₙ) and v = (v₁, v₂, ..., vₙ), then u + v = (u₁+v₁,
u₂+v₂, ..., uₙ+vₙ).
4. Q: What is scalar multiplication of a vector? ANSWER Multiplying a
vector by a scalar c multiplies each component by c: c·(v₁, v₂, ..., vₙ) = (cv₁, cv₂,
..., cvₙ).
5. Q: What is the zero vector? ANSWER The zero vector is the vector with
all components equal to zero, denoted 0 or 0. It acts as the additive identity.
6. Q: What is the dot product (inner product) of two vectors? ANSWER
For vectors u and v in ℝⁿ, u·v = u₁v₁ + u₂v₂ + ... + uₙvₙ. It produces a scalar.
7. Q: What does it mean for two vectors to be orthogonal? ANSWER Two
vectors are orthogonal if their dot product equals zero: u·v = 0.
8. Q: What is the magnitude (norm) of a vector? ANSWER The magnitude
of vector v is ||v|| = √(v₁² + v₂² + ... + vₙ²).
9. Q: What is a unit vector? ANSWER A unit vector is a vector with
magnitude 1. Any vector can be normalized to a unit vector by dividing by its
magnitude.
10. Q: What is a vector space? ANSWER A vector space is a set V with
operations of addition and scalar multiplication satisfying eight axioms (closure,
associativity, commutativity, identity elements, inverses, and distributivity).
11. Q: What is a subspace? ANSWER A subspace is a subset of a vector
space that is itself a vector space under the same operations. It must contain the
zero vector and be closed under addition and scalar multiplication.
, 12. Q: What is the span of a set of vectors? ANSWER The span is the set of
all possible linear combinations of those vectors. It forms a subspace.
13. Q: What does it mean for vectors to be linearly independent?
ANSWER Vectors are linearly independent if no vector can be written as a
linear combination of the others. Equivalently, c₁v₁ + ... + cₙvₙ = 0 implies all cᵢ
= 0.
14. Q: What does it mean for vectors to be linearly dependent? ANSWER
Vectors are linearly dependent if at least one can be written as a linear
combination of the others, or if c₁v₁ + ... + cₙvₙ = 0 for some non-zero
coefficients.
15. Q: What is a basis for a vector space? ANSWER A basis is a linearly
independent set of vectors that spans the entire vector space.
16. Q: What is the dimension of a vector space? ANSWER The dimension is
the number of vectors in any basis for the space. All bases have the same
number of vectors.
17. Q: What is the standard basis for ℝⁿ? ANSWER The standard basis
consists of vectors e₁ = (1,0,...,0), e₂ = (0,1,...,0), etc., with a 1 in one position
and 0s elsewhere.
18. Q: What is the cross product of two vectors in ℝ³? ANSWER For u =
(u₁,u₂,u₃) and v = (v₁,v₂,v₃), u×v = (u₂v₃-u₃v₂, u₃v₁-u₁v₃, u₁v₂-u₂v₁). The result is
orthogonal to both vectors.
19. Q: What is the projection of vector u onto vector v? ANSWER
proj_v(u) = ((u·v)/(v·v))v. This gives the component of u in the direction of v.
20. Q: What is the angle between two vectors? ANSWER The angle θ
between vectors u and v satisfies cos(θ) = (u·v)/(||u||·||v||).
21. Q: What is a linear combination? ANSWER A linear combination of
vectors v₁, ..., vₙ is an expression c₁v₁ + c₂v₂ + ... + cₙvₙ where c₁, ..., cₙ are
scalars.
22. Q: What is the coordinate vector of v relative to a basis B? ANSWER
It's the vector of coefficients when v is written as a linear combination of the
basis vectors.
23. Q: Can a set containing the zero vector be linearly independent?
ANSWER No, any set containing the zero vector is linearly dependent because
1·0 = 0.
QUESTIONS AND ANSWERS - CURRENT VERSION (2026/2027)
1. Q: What is a vector? ANSWER A vector is a mathematical object with
both magnitude and direction, typically represented as an ordered list of
numbers (components) in n-dimensional space.
2. Q: What is the difference between a row vector and a column vector?
ANSWER A row vector is written horizontally as [a₁, a₂, ..., aₙ], while a column
vector is written vertically. They are transposes of each other.
3. Q: How do you add two vectors? ANSWER Add corresponding
components: if u = (u₁, u₂, ..., uₙ) and v = (v₁, v₂, ..., vₙ), then u + v = (u₁+v₁,
u₂+v₂, ..., uₙ+vₙ).
4. Q: What is scalar multiplication of a vector? ANSWER Multiplying a
vector by a scalar c multiplies each component by c: c·(v₁, v₂, ..., vₙ) = (cv₁, cv₂,
..., cvₙ).
5. Q: What is the zero vector? ANSWER The zero vector is the vector with
all components equal to zero, denoted 0 or 0. It acts as the additive identity.
6. Q: What is the dot product (inner product) of two vectors? ANSWER
For vectors u and v in ℝⁿ, u·v = u₁v₁ + u₂v₂ + ... + uₙvₙ. It produces a scalar.
7. Q: What does it mean for two vectors to be orthogonal? ANSWER Two
vectors are orthogonal if their dot product equals zero: u·v = 0.
8. Q: What is the magnitude (norm) of a vector? ANSWER The magnitude
of vector v is ||v|| = √(v₁² + v₂² + ... + vₙ²).
9. Q: What is a unit vector? ANSWER A unit vector is a vector with
magnitude 1. Any vector can be normalized to a unit vector by dividing by its
magnitude.
10. Q: What is a vector space? ANSWER A vector space is a set V with
operations of addition and scalar multiplication satisfying eight axioms (closure,
associativity, commutativity, identity elements, inverses, and distributivity).
11. Q: What is a subspace? ANSWER A subspace is a subset of a vector
space that is itself a vector space under the same operations. It must contain the
zero vector and be closed under addition and scalar multiplication.
, 12. Q: What is the span of a set of vectors? ANSWER The span is the set of
all possible linear combinations of those vectors. It forms a subspace.
13. Q: What does it mean for vectors to be linearly independent?
ANSWER Vectors are linearly independent if no vector can be written as a
linear combination of the others. Equivalently, c₁v₁ + ... + cₙvₙ = 0 implies all cᵢ
= 0.
14. Q: What does it mean for vectors to be linearly dependent? ANSWER
Vectors are linearly dependent if at least one can be written as a linear
combination of the others, or if c₁v₁ + ... + cₙvₙ = 0 for some non-zero
coefficients.
15. Q: What is a basis for a vector space? ANSWER A basis is a linearly
independent set of vectors that spans the entire vector space.
16. Q: What is the dimension of a vector space? ANSWER The dimension is
the number of vectors in any basis for the space. All bases have the same
number of vectors.
17. Q: What is the standard basis for ℝⁿ? ANSWER The standard basis
consists of vectors e₁ = (1,0,...,0), e₂ = (0,1,...,0), etc., with a 1 in one position
and 0s elsewhere.
18. Q: What is the cross product of two vectors in ℝ³? ANSWER For u =
(u₁,u₂,u₃) and v = (v₁,v₂,v₃), u×v = (u₂v₃-u₃v₂, u₃v₁-u₁v₃, u₁v₂-u₂v₁). The result is
orthogonal to both vectors.
19. Q: What is the projection of vector u onto vector v? ANSWER
proj_v(u) = ((u·v)/(v·v))v. This gives the component of u in the direction of v.
20. Q: What is the angle between two vectors? ANSWER The angle θ
between vectors u and v satisfies cos(θ) = (u·v)/(||u||·||v||).
21. Q: What is a linear combination? ANSWER A linear combination of
vectors v₁, ..., vₙ is an expression c₁v₁ + c₂v₂ + ... + cₙvₙ where c₁, ..., cₙ are
scalars.
22. Q: What is the coordinate vector of v relative to a basis B? ANSWER
It's the vector of coefficients when v is written as a linear combination of the
basis vectors.
23. Q: Can a set containing the zero vector be linearly independent?
ANSWER No, any set containing the zero vector is linearly dependent because
1·0 = 0.
