# 5: Graphical Explorations of Vectors

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We already looked at the basics of graphing vectors. In this chapter, we’ll explore these ideas more fully. One often gains a better understanding of a concept by “seeing” it. For instance, one can study the function $$f(x) = x^{2}$$ and describe many properties of how the output relates to the input without producing a graph, but the graph can quickly bring meaning and insight to equations and formulae. Not only that, but the study of graphs of functions is in itself a wonderful mathematical world, worthy of exploration.

We’ve studied the graphing of vectors; in this chapter we’ll take this a step further and study some fantastic graphical properties of vectors and matrix arithmetic. We mentioned earlier that these concepts form the basis of computer graphics; in this chapter, we’ll see even better how that is true.

Thumbnail: A linear combination of one basis set of vectors (purple) obtains new vectors (red). If they are linearly independent, these form a new basis set. The linear combinations relating the first set to the other extend to a linear transformation, called the change of basis. (CC0; Maschen via Wikipedia)

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