Linear Algebra: Unit 1

MODULE 1

What is a scalar?

A single real number used for measurement (element of ℝ).

What is a vector?

An ordered list of scalars (element of ℝⁿ).

How do you notate a vector?

Horizontally: ⟨x₁, x₂, …, xₙ⟩; Vertically: column format with an arrow.

What is ℝⁿ?

The space of all vectors with n real-number entries (n-dimensional Euclidean space).

What is the formula for the magnitude of a vector?

‖v‖ = √(v₁² + v₂² + … + vₙ²)

How do you visualize a vector in ℝ²?

As an arrow from the origin to point (x, y) on the xy-plane.

Give an example of a vector in ℝ³

⟨3, 3, -4⟩, which can be visualized in 3D space.

Are ∫ and Σ scalars?

No, they are operations, not elements of ℝ.

How do you add two vectors?

Add corresponding coordinates: v + w = ⟨v₁ + w₁, …, vₙ + wₙ⟩

When is vector addition valid?

Only when the two vectors have the same number of coordinates (same dimension).

What is scalar-vector multiplication?

Each coordinate of the vector is multiplied by the scalar: c·v = ⟨c·v₁, …, c·vₙ⟩

What does scalar-vector multiplication do geometrically?

Stretches vector by |c|, reverses direction if c < 0.

How is vector subtraction defined?

v - w = v + (-1)·w

How do you compute distance between two vectors?

Distance = ‖v - w‖

What is a unit vector?

A vector with length (magnitude) 1.

How do you normalize a vector v?

v̂ = (1 / ‖v‖) · v

What is a linear combination?

An expression like c₁v₁ + c₂v₂ + … + cₖvₖ where the c's are scalars.

What is the span of a set of vectors?

All vectors that can be formed as linear combinations of the set.

What is the dot product of two vectors?

v · w = v₁w₁ + v₂w₂ + … + vₙwₙ

What does the dot product output?

A scalar (not a vector).

How is vector length related to dot product?

‖v‖² = v · v

How is dot product used to find angle between vectors?

v · w = ‖v‖‖w‖cos(θ)

How to tell if two vectors are orthogonal?

If v · w = 0, they are orthogonal (perpendicular).

What is a vector equation?

An equation of the form x₁·v₁ + x₂·v₂ + … + xₖ·vₖ = b, where v₁,…,vₖ are known vectors and x₁,…,xₖ are unknown scalars.

When does a vector equation have a solution?

When the vector b is in the span of v₁,…,vₖ.

What does it mean if b ∉ span{v₁,…,vₖ}?

There is no solution to the vector equation.

How is solving a vector equation related to linear systems?

Solving x₁·v₁ + … + xₖ·vₖ = b is equivalent to solving the system Ax = b.

What are the conditions for a unique solution to Ax = b?

b must be in the span of A's columns, and the columns must be linearly independent.

What happens if the columns of A are linearly dependent?

If b is in the span, there will be infinitely many solutions.

What is the geometric interpretation of span{v₁,…,vₖ}?

It forms a k-dimensional subspace through the origin in ℝⁿ.

What does the span of one vector look like?

A line through the origin.

What does the span of two vectors look like in ℝ³?

A plane through the origin, if they are not parallel.

How are vector equations used in data science?

They model predictions, regressions, and transformations using data matrices.