Linear combinations are really important for understanding the size of vector spaces. The size, or dimension, of a vector space tells us the most number of vectors that can’t be made from each other.

1. What’s a Linear Combination?: A vector, which we can call $\mathbf{v}$, is a linear combination of other vectors $\{\mathbf{u}_1, \mathbf{u}_2, ..., \mathbf{u}_n\}$ if we can write it using some numbers, called scalars, like this:

   $$\mathbf{v} = c_1\mathbf{u}_1 + c_2\mathbf{u}_2 + ... + c_n\mathbf{u}_n$$

2. What is a Span?: When we take all possible linear combinations of a specific set of vectors $\{ \mathbf{u}_1, ..., \mathbf{u}_n \}$, we create something called a span. We write it as $Span(\{\mathbf{u}_1, ..., \mathbf{u}_n\})$.

3. Basis and Dimension: A basis is a special group of vectors that are all different from each other and can represent the whole space. The dimension, or size, which we call $d$, is just the number of vectors in any basis:

   $$\text{dim}(\mathbf{V}) = d$$

This means that linear combinations help us see how vectors connect to the overall shape and size of vector spaces.