Standard basis of r4

The vector space R4 has basis B = 1 e1, e2, e3 + e2, e4 + e1l. Give a brief proof. Solution note: There are four vectors, so by Theorem 3.3.

1. What is a basis for R4?
2. What is standard basis for R3?
3. What is standard ordered basis for R2?
4. What is the dimension of R4?

What is a basis for R4?

A basis for R4 always consists of 4 vectors. (TRUE: Vectors in a basis must be linearly independent AND span.) 4. The union of two subspaces is a subspace.

What is standard basis for R3?

A basis of R3 cannot have more than 3 vectors, because any set of 4 or more vectors in R3 is linearly dependent. A basis of R3 cannot have less than 3 vectors, because 2 vectors span at most a plane (challenge: can you think of an argument that is more “rigorous”?).

What is standard ordered basis for R2?

The standard ordered basis of ℝ2 is e1, e2. Let T : ℝ2 → ℝ2 be the linear transformation such that T reflects the points through the line x1 = -x2.

What is the dimension of R4?

The space R4 is four-dimensional, and so is the space M of 2 by 2 matrices. Vectors in those spaces are determined by four numbers. The solution space Y is two-dimensional, because second order differential equations have two independent solutions.