Linear Algebra/Definition and Examples of Vector Spaces

The best way to go through the examples below is to check all ten conditions in the definition. That check is written out at length in the first example. Use it as a model for the others. Especially important are the first condition "$$\vec v+\vec w$$ is in $$ V $$" and the sixth condition "$$r\cdot\vec v$$ is in $$V$$". These are the closure conditions. They specify that the addition and scalar multiplication operations are always sensible — they are defined for every pair of vectors, and every scalar and vector, and the result of the operation is a member of the set (see Example 1.4).

In a similar way, each $$ \mathbb{R}^n $$ is a vector space with the usual operations of vector addition and scalar multiplication. (In $$ \mathbb{R}^1 $$, we usually do not write the members as column vectors, i.e., we usually do not write "$$ (\pi) $$". Instead we just write "$$ \pi $$".)

A vector space must have at least one element, its zero vector. Thus a one-element vector space is the smallest one possible.

Examples
As we've done in those equations, we often omit the multiplication symbol "$$ \cdot $$". We can distinguish the multiplication in "$$ c_1v_1 $$" from that in "$$ r\vec{v}\, $$" since if both multiplicands are real numbers then real-real multiplication must be meant, while if one is a vector then scalar-vector multiplication must be meant.

The prior example has brought us full circle since it is one of our motivating examples.

Summary
We finish with a recap.

Our study in Chapter One of Gaussian reduction led us to consider collections of linear combinations. So in this chapter we have defined a vector space to be a structure in which we can form such combinations, expressions of the form $$ c_1\cdot\vec{v}_1+\dots+c_n\cdot\vec{v}_n $$ (subject to simple conditions on the addition and scalar multiplication operations). In a phrase: vector spaces are the right context in which to study linearity.

Finally, a comment. From the fact that it forms a whole chapter, and especially because that chapter is the first one, a reader could come to think that the study of linear systems is our purpose. The truth is, we will not so much use vector spaces in the study of linear systems as we will instead have linear systems start us on the study of vector spaces. The wide variety of examples from this subsection shows that the study of vector spaces is interesting and important in its own right, aside from how it helps us understand linear systems. Linear systems won't go away. But from now on our primary objects of study will be vector spaces.

Exercises
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