8.3 VectorSpaces and Subspaces

[Pages:5]8.3 Vector Spaces and Subspaces

Performance Criterion: 8. (d) Determine whether a subset of Rn is a subspace. If so, prove it; if not, give an appropriate counterexample.

Vector Spaces

The term "space" in math simply means a set of objects with some additional special properties. There are metric spaces, function space, topological spaces, Banach spaces, and more. The vectors that we have been dealing with make up the vector spaces called R2, R3 and, for larger values, Rn. In general, a vector space is simply a collection of objects called vectors (and a set of scalars) that satisfy certain properties.

Definition 8.3.1: Vector Space A vector space is a set V of objects called vectors and a set of scalars (usually the real numbers R), with the operations of vector addition and scalar multiplication, for which the following properties hold for all u, v, w in V and scalars c and d.

1. u + v is in V 2. u + v = v + u 3. (u + v) + w = u + (v + w) 4. There exists a vector 0 in V such that u + 0 = u. This vector is called the

zero vector. 5. For every u in V there exists a vector -u in V such that u + (-u) = 0. 6. cu is in V . 7. c(u + v) = cu + cv 8. (c + d)u = cu + du 9. c(du) = (cd)u 10. 1u = u

Note that items 1 and 6 of the above definition say that the vector space V is closed under addition and scalar multiplication.

When working with vector spaces, we will be very interested in certain subsets of those vector spaces that are the span of a set of vectors. As you proceed, recall Example 8.2(b), where we showed that the span of a set of vectors is closed under addition and scalar multiplication.

Subspaces of Vector Spaces

As you should know by now, the two main operations with vectors are multiplication by scalars and addition of vectors. (Note that these two combined give us linear combinations, the foundation of almost everything we've done.) A given vector space can have all sorts of subsets; consider the following subsets of R2.

? The set S1 consisting of the first quadrant and the nonnegative parts of the two axes, or all vectors of the

form

x1 x2

such that x1 0 and x2 0.

110

? The set S2 consisting of the line containing the vector

t

3 2

where t ranges over all real numbers.

3 2

.

Algebraically this is all vectors of the form

? The set S3 consisting of the first and third quadrants and both axes. This can be described as the set of all

vectors

x1 x2

with x1x2 0.

Our current concern is whether these subsets of R2 are closed under addition and scalar multiplication. With a bit of thought you should see that S1 is closed under addition, but not scalar multiplication when the scalar is negative:

S1

u+v

S1

u

w

v

cw c ................
................

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