Questions: Vectors in R^n

Question 1

A machine learning engineer represents a 28×28 grayscale image as a vector in ℝ⁷⁸⁴. She applies a linear algebra theorem proved for arbitrary ℝⁿ to analyze this data. What justifies applying that theorem here?

• A. The theorem was probably derived with image data in mind, making it specifically applicable
• B. She needs to re-derive the theorem for ℝ⁷⁸⁴ because theorems in lower dimensions don't generalize automatically
• C. Because algebraic operations on vectors work identically in any ℝⁿ, theorems proved for arbitrary n apply immediately to ℝ⁷⁸⁴
• D. ℝ⁷⁸⁴ is small enough to be handled by geometric intuition from ℝ³

Answer: (C)

This is the power of abstraction in linear algebra. Addition, scalar multiplication, dot products, and the theorems built from them work identically regardless of dimension. A theorem proved for arbitrary ℝⁿ — say, about linear independence or orthogonality — applies to ℝ⁷⁸⁴ without modification. The engineer does not need to re-derive anything; the abstract framework handles all dimensions at once. This is why the move from ℝ³ to ℝⁿ is mathematically effortless once you accept it algebraically.

Question 2

What distinguishes the zero vector from all other vectors in ℝⁿ?

• A. It has zero magnitude and points in the negative direction
• B. It has zero magnitude and no defined direction
• C. It is not a valid vector because it carries no information
• D. It has zero direction but a defined magnitude of 1

Answer: (B)

The zero vector ⟨0, 0, ..., 0⟩ has magnitude zero (‖0‖ = 0), but direction is undefined for it — there is no 'which way it points.' Every nonzero vector can be normalized to a unit vector that preserves direction, but the zero vector cannot be normalized because dividing by zero is undefined. This distinguishes it from all other vectors, which carry both magnitude and direction.

Question 3

A vector in ℝ² and a point in the 2D coordinate plane contain different types of mathematical information.

• T. True
• F. False

Answer: False

A 2D vector ⟨3, −1⟩ and the point (3, −1) contain identical mathematical data — the same two numbers in the same order. The distinction is one of interpretation: a point emphasizes location, while a vector emphasizes displacement or direction. But as mathematical objects, they are the same ordered pair. This equivalence extends to ℝⁿ: a data point with n measurements and a vector in ℝⁿ are the same object interpreted differently.

Question 4

The magnitude of a vector in ℝⁿ is computed by taking the square root of the sum of the squares of its components, which is a direct generalization of the Pythagorean theorem.

• T. True
• F. False

Answer: True

‖v‖ = √(v₁² + v₂² + ... + vₙ²) is exactly the Pythagorean theorem extended to n dimensions. In ℝ², this gives the familiar distance formula: the length of the hypotenuse is √(a² + b²). In ℝ³, you apply Pythagoras twice (once in the base plane, once for the vertical component). The formula in ℝⁿ continues the same pattern, making it a genuine generalization of the 2D result.

Question 5

Why is writing vectors as columns (rather than rows) a useful convention when working with matrix-vector multiplication?

Model answer: When a matrix A multiplies a column vector v, the result is a linear combination of A's columns: the first column of A scaled by v₁, plus the second column scaled by v₂, and so on. This column-combination interpretation is natural and geometric — it shows what the matrix is doing to the vector. If vectors were written as rows, this intuition would break down, requiring transpositions to recover it. The column convention aligns the operation's notation with its geometric meaning.

The column convention isn't arbitrary — it matches how matrix-vector multiplication is defined. With column vectors, Av = v₁(column 1 of A) + v₂(column 2 of A) + ... + vₙ(column n of A). This lets you interpret a matrix as a set of basis vectors being scaled and combined, which is the geometric heart of linear transformations. The convention pays dividends throughout linear algebra, particularly when you study linear transformations, change of basis, and eigendecomposition.