Week 1 — Vector Space Structure of R^n

Real and Complex Vectors

An n-dimensional real vector is an n-tuple (x_1,\ldots, x_n) where the x_i are real numbers. If the x_i are complex instead of real, we have an n-dimensional complex vector.

We write \mathbf{R}^{n} and \mathbf{C}^{n} for the sets of n-dimensional real and complex vectors respectively.

One can visualize an element x=(x_1,\ldots,x_n) of \mathbf{R}^{n} as an “arrow” with its base at the origin and its endpoint at the point (x_1,\ldots, x_n) in n-dimensional Euclidean space.