Page 102 - 35Linear Algebra

102 Vector Spaces

(+iv) (Zero) There is a special vector 0 V ∈ V such that u + 0 V = u for all u
in V .

(+v) (Additive Inverse) For every u ∈ V there exists w ∈ V such that
u + w = 0 V .

(· i) (Multiplicative Closure) c · v ∈ V . Scalar times a vector is a vector.

(· ii) (Distributivity) (c+d)·v = c·v +d·v. Scalar multiplication distributes
over addition of scalars.

(· iii) (Distributivity) c·(u+v) = c·u+c·v. Scalar multiplication distributes
over addition of vectors.

(· iv) (Associativity) (cd) · v = c · (d · v).

(· v) (Unity) 1 · v = v for all v ∈ V .

Examples of each rule

Remark Rather than writing (V, +, . , R), we will often say “let V be a vector space
over R”. If it is obvious that the numbers used are real numbers, then “let V be a
vector space” suffices. Also, don’t confuse the scalar product · with the dot product .
The scalar product is a function that takes as its two inputs one number and one
vector and returns a vector as its output. This can be written

·: R × V → V .

Similarly
+ : V × V → V .

On the other hand, the dot product takes two vectors and returns a number. Suc-
cinctly: : V × V → R. Once the properties of a vector space have been verified,
we’ll just write scalar multiplication with juxtaposition cv = c · v, though, to keep our
notation efficient.

5.1 Examples of Vector Spaces

One can find many interesting vector spaces, such as the following:

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