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linear algebra specifically the 2nd chapter of sheldon axler's linear algebra done right

Welcome! I'm excited to dive into Sheldon Axler's classic text with you. Chapter 2 is really where the abstract beauty of linear algebra begins to take shape.

Before we start exploring concepts like span and linear independence, I'd love to learn a little bit about your mathematical background and what you hope to get out of our sessions.

I've finished Chapter 1 and am self-studying for fun.

That is wonderful! Self-studying for sheer curiosity is the absolute best way to learn, and since you have Chapter 1 under your belt, you are perfectly primed for the beautiful abstractions of Chapter 2.

Your path
Bases
Dimension
Let's kick things off with the building blocks of the entire chapter: and .
Imagine you are standing at the origin in 2D space, and you are only allowed to walk in two directions: one step along the vector v1=(1,0)v_1 = (1, 0) or one step along the vector v2=(0,1)v_2 = (0, 1). By scaling these steps and adding them together, you can reach any point in the plane. For example, to reach the point (3,−2)(3, -2), you would compute the linear combination: 3(1,0)−2(0,1)=(3,−2)3(1, 0) - 2(0, 1) = (3, -2) Because every single vector in R2\mathbb{R}^2 can be reached this way, we say that the list (1,0),(0,1)(1, 0), (0, 1) spans R2\mathbb{R}^2.

To make sure this feels intuitive before we introduce the formal definitions, let me ask you: if we only had the single vector (1,1)(1, 1) in R2\mathbb{R}^2, what would its span look like geometrically?