Vector spaces linear algebra

Because it involves two kinds of addition and two kinds of multiplication, that definition may seem confused. These expressions aren't ambiguous because, e. The best way to go through the examples below is to check all ten conditions in the definition.

Vector space definition and examples

That check is written out at length in the first example. Use it as a model for the others. 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.

There are five conditions in item 1. For 2, that addition of vectors commutes, take all entries to be real numbers and compute. Condition 3, associativity of vector addition, is similar. The checks for the five conditions having to do with scalar multiplication are just as routine. Next, this checks 7. Of course, this example of closure is not a proof of closure.

Vector spaces linear algebra pdf

Thus the two closure conditions are satisfied. Verification of the other conditions in the definition of a vector space are just as straightforward. Example 1. In contrast with those two, consider the set of two-tall columns with entries that are integers under the obvious operations.