3.3. Linear Combinations, Independence, Basis, and Dimension

This section introduces some important foundational concepts that will be used in later sections.

3.3.1. Linear Combinations

Consider a set of k real-valued variables x₁, x₂,..., x_k. Suppose that we are given a set of k real-valued weights w₁, w₂,..., w_k. Then, we can define the weighted sum of these variables as s = w₁x₁ + w₂x₂ +...+ w_kx_k. This sum “mixes” the variables in linear proportion to the weights. Therefore, we call s a linear combination of the variables.

We can generalize the notion of linear combination to vectors. Here, each x_i is a vector, so that their linear combination, s, is also a vector. Of course, each vector must have the same number of elements. Note that each component of s is a linear combination of the corresponding elements of the underlying vectors.