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Section 1.5 Bases

Intuitively, vectors in space can point up or down, right or left, and forward or backward. “Space” in this case refers to our ordinary notion of the Euclidean geometry which we experience in our daily lives. Such vectors are said to be three dimensional. As a special case, some vectors are restricted to a plane, and are said to be two dimensional.

A two-dimensional vector, such as the one shown in Figure 1.6, can be expressed in terms of how much it goes “to the right” in the direction of \(\aa\) and how much it goes “up” in the direction of \(\bb,\) as shown in Figure 1.9. This process is called expanding the vector in terms of the basis vectors \(\aa\) and \(\bb\text{.}\)

Figure 1.9. A two-dimensional vector \(\ww\text{,}\) expanded in terms of a basis \(\{\aa,\bb\}\text{.}\)

There is no requirement that the basis vectors \(\aa\) and \(\bb\) used in the expansion be “perpendicular”, nor that they have the same “magnitude”. In fact, we do not yet know what those words mean (see Section 1.7)! Nonetheless, we can conclude from Figure 1.9 that \(\ww=2\aa+\bb\text{.}\)

In two dimensions, it takes two basis vectors in order to expand any vector. Any two linearly-independent (that is, non-parallel) vectors in the plane can be used to expand all vectors in the plane. In three dimensions, it takes a third basis vector, that is not in the plane defined by the other two, to expand any vector.