LINEAR ALGEBRA TOP STUDY GUIDE FINAL
EXAM 2026 COMPLETE QUESTIONS
ANSWERS PREMIUM STUDY GUIDE
◉ 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.
◉ What is the vector equation of a line through point P0?
Answer: x = P0 + td, where t ∈ R and d is the direction vector.
◉ What is the equation of a plane in R3?
Answer: n · (x - P0) = 0, where n is the normal vector.
◉ How do you find the normal vector given two vectors in the plane?
Answer: n = u × v (cross product).
◉ What is the distance from a point Q to a line in R3?
Answer: D = ∥(Q - P0) × d∥ / ∥d∥.
◉ What is a linear system of equations?
Answer: A system of m linear equations in n unknowns has the form
Ax = b.
◉ What are the possible outcomes of a linear system?
EXAM 2026 COMPLETE QUESTIONS
ANSWERS PREMIUM STUDY GUIDE
◉ 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.
◉ What is the vector equation of a line through point P0?
Answer: x = P0 + td, where t ∈ R and d is the direction vector.
◉ What is the equation of a plane in R3?
Answer: n · (x - P0) = 0, where n is the normal vector.
◉ How do you find the normal vector given two vectors in the plane?
Answer: n = u × v (cross product).
◉ What is the distance from a point Q to a line in R3?
Answer: D = ∥(Q - P0) × d∥ / ∥d∥.
◉ What is a linear system of equations?
Answer: A system of m linear equations in n unknowns has the form
Ax = b.
◉ What are the possible outcomes of a linear system?
