LINEAR ALGEBRA TOP STUDY GUIDE ACTUAL
TEST PAPER 2026 QUESTIONS SOLUTIONS
GRADED A+
◉ What is the condition for two vectors to be equal?
Answer: Two vectors are equal if all corresponding components are
equal.
◉ What is the formula for vector addition?
Answer: u + v = [u1 + v1, u2 + v2, ..., un + vn]⊤
◉ How is scalar multiplication of a vector defined?
Answer: cv = [cv1, cv2, ..., cvn]⊤
◉ What is the zero vector in Rn?
Answer: 0 = [0, 0, ..., 0]⊤
◉ What does the parallelogram law relate to?
Answer: Vector addition geometrically relates to the parallelogram
law.
◉ What is the dot product of two vectors u and v?
, Answer: u · v = u1v1 + u2v2 + ... + unvn = u⊤v
◉ How is the norm (length) of a vector v defined?
Answer: ∥v∥ = √(v1² + v2² + ... + vn²)
◉ What is the definition of a unit vector?
Answer: A unit vector ˆv = v / ∥v∥, normalizing v to length 1.
◉ What theorem relates the angle between two vectors?
Answer: For nonzero u, v ∈Rn, cos θ = (u · v) / (∥u∥∥v∥).
◉ What does it mean for vectors u and v to be orthogonal?
Answer: u and v are orthogonal if u · v = 0.
◉ What is the Cauchy-Schwarz inequality?
Answer: |u · v| ≤ ∥u∥∥v∥.
◉ What is the triangle inequality for vectors?
Answer: ∥u + v∥ ≤ ∥u∥ + ∥v∥.
◉ What is the vector projection of v onto u?
Answer: proju(v) = (u · v / (u · u)) u.
TEST PAPER 2026 QUESTIONS SOLUTIONS
GRADED A+
◉ What is the condition for two vectors to be equal?
Answer: Two vectors are equal if all corresponding components are
equal.
◉ What is the formula for vector addition?
Answer: u + v = [u1 + v1, u2 + v2, ..., un + vn]⊤
◉ How is scalar multiplication of a vector defined?
Answer: cv = [cv1, cv2, ..., cvn]⊤
◉ What is the zero vector in Rn?
Answer: 0 = [0, 0, ..., 0]⊤
◉ What does the parallelogram law relate to?
Answer: Vector addition geometrically relates to the parallelogram
law.
◉ What is the dot product of two vectors u and v?
, Answer: u · v = u1v1 + u2v2 + ... + unvn = u⊤v
◉ How is the norm (length) of a vector v defined?
Answer: ∥v∥ = √(v1² + v2² + ... + vn²)
◉ What is the definition of a unit vector?
Answer: A unit vector ˆv = v / ∥v∥, normalizing v to length 1.
◉ What theorem relates the angle between two vectors?
Answer: For nonzero u, v ∈Rn, cos θ = (u · v) / (∥u∥∥v∥).
◉ What does it mean for vectors u and v to be orthogonal?
Answer: u and v are orthogonal if u · v = 0.
◉ What is the Cauchy-Schwarz inequality?
Answer: |u · v| ≤ ∥u∥∥v∥.
◉ What is the triangle inequality for vectors?
Answer: ∥u + v∥ ≤ ∥u∥ + ∥v∥.
◉ What is the vector projection of v onto u?
Answer: proju(v) = (u · v / (u · u)) u.
