Introduction to Vector Spaces

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A vector space (over ) consists of a set along with two operations "" and "" subject to these conditions.

  1. For any .
  2. For any .
  3. For any .
  4. There is a zero vector such that for all .
  5. Each has an additive inverse such that .
  6. If is a scalar, that is, a member of and then the scalar multiple is in .
  7. If and then .
  8. If and , then .
  9. If and , then .
  10. For any , .

Remark: Because it involves two kinds of addition and two kinds of multiplication, that definition may seem confused. For instance, in condition 7 "", the first "" is the real number addition operator while the "" to the right of the equals sign represents vector addition in the structure . These expressions aren't ambiguous because, e.g., and are real numbers so "" can only mean real number addition.

Lemma 1.17: In any vector space , for any and , we have

  1. , and
  2. , and
  3. .

Proof: For 1, note that . Add to both sides the additive inverse of , the vector such that .

The second item is easy: shows that we can write "" for the additive inverse of without worrying about possible confusion with .

For 3, this will do.


Example 1

The set is a vector space if the operations "" and "" have their usual meaning.

We shall check all of the conditions.

There are five conditions in item 1. For 1, closure of addition, note that for any the result of the sum

is a column array with two real entries, and so is in . For 2, that addition of vectors commutes, take all entries to be real numbers and compute

(the second equality follows from the fact that the components of the vectors are real numbers, and the addition of real numbers is commutative). Condition 3, associativity of vector addition, is similar.

For the fourth condition we must produce a zero element — the vector of zeroes is it.

For 5, to produce an additive inverse, note that for any we have

so the first vector is the desired additive inverse of the second.

The checks for the five conditions having to do with scalar multiplication are just as routine. For 6, closure under scalar multiplication, where ,

is a column array with two real entries, and so is in . Next, this checks 7.

For 8, that scalar multiplication distributes from the left over vector addition, we have this.

The ninth

and tenth conditions are also straightforward.

In a similar way, each is a vector space with the usual operations of vector addition and scalar multiplication. (In , we usually do not write the members as column vectors, i.e., we usually do not write "". Instead we just write "".)


Example 2

The set of real-valued functions of the real variable is a vector space under the operations

and

inherited from the space in the prior example. (We can think of as "the same" as in that corresponds to the vector with components and .)