Foundations and Geometry

Why study ℝ² and ℝ³ first?

To build geometric intuition for abstract spaces.

When is u · v = 0?

When vectors are orthogonal.

What does ‖u × v‖ represent?

Area of parallelogram spanned by u and v.

What does r(t) = r₀ + tv represent?

A line as points along a direction vector.

What is a normal vector to a hyperplane?

A vector orthogonal to the hyperplane.

Why identify vectors with points in ℝⁿ?

To view linear maps as geometric transformations.

What does distance from point to line measure?

Distance from the subspace.

Equipollent segments ensure what?

Each vector corresponds to a unique point.