Section 3.1

Vector Spaces

Vector-Space Operations

In Euclidean space R^n we have,

1.) Addition of two vectors v, u in R^n is again in R^n . (Closed under vector addition.)

2.) Multiplication of a vector v in R^n with a scalar alpha is again in R^n . (Closed under scalar multiplication.)

We have encountered subsets in R^n that are closed under addition and multiplication. We called

these sets subspaces . ( Section 1.6)

>

> `S2 ={[x,y,z] | 2*x+3*y+z=0}`;

`S2 ={[x,y,z] | 2*x+3*y+z=0}`

>

Consider the set C[a,b] of continuous functions on an interval [a,b]. This set is closed

under addition and scalar multiplication. Can we make a general definition and not

restrict ourselves to Euclidean spaces R^n .

Definition of a Vector Space

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A (real) vector space is a set V of objects called vectors , together with a rule for

adding any two vectors v and w to produce v+w in V and a rule for multiplying

any vector v in V by any scalar r in R to produce a vector rv in V. Moreover, there

must exist a vector 0 in V and for each v in V there must exist a vector -v in V such

that properties A1-A4 and S1-S4 hold for vectors v, w, u and scalars r,s,

A1: (u + v) + w = u + (v + w)

A2: v + w = w + v

A3: 0 + v = v

A4: v + (-v) = 0

S1: r(v + w) = rv + rw

S2: (r + s)v = rv + sv

S3: r(sv) = (rs)v

S4: 1v = v

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A "vector" in a general vector space does not always mean an element with a specified

number of components. It may refer to a function if we are dealing with a set of functions.

Many examples of such spaces are studied in precalculus, calculus,and differential equation

courses.

Examples of Vector Spaces

Example 1: Set R^n of all n x 1 vectors is a vector space, using the usual

vector addition and scalar multiplication.

Example 2: Set M[m,n] of all m x n matrices is a vector space, using as vector

addition and scalar multiplication the usual addition of matrices and multiplication

of a matrix by a scalar.

Example 3: Set P[n] of all polynomials of degree <= n. i.e.

p(x) = a[n]*x^n+a[n-1]*x^`n-1` + . . . + a[1]*x+a[0] with the usual addition and

scalar multiplication i.e.,

(p+q)(x) = p(x) + q(x) for p and q in P[n]

(rp)(x) = rp(x) for p in P[n] and r a scalar

is a vector space.

Example 4: Set C[a,b] of all continous real-valued functions on [a,b] with

the usual addition and scalar multiplication i.e.,

(f+g)(x) = f(x) + g(x) for f and g in C[a,b]

(rf)(x) = rf(x) for f in C[a,b] and r a scalar

is a vector space.

Example 5: Set S of all infinite sequences { a[n] } of real numbers with scalar

multiplication r{ a[n] } = {r a[n] } and addition { a[n] } + { b[n] } = { a[n]+b[n] } is

a vector space.

More examples

Example 6: Let S be the set of real numbers with an unsual definition of addition,

(x + y) = max{x,y} and usual scalar multiplication (rx) = rx. Is S with these

operations a Vector Space?

Must show closure and properties A1-A4 and S1-S4.

A1: (u + v) + w = u + (v + w)

A2: v + w = w + v

A3: 0 + v = v

A4: v + (-v) = 0

S1: r(v + w) = rv + rw

S2: (r + s)v = rv + sv

S3: r(sv) = (rs)v

S4: 1v = v

Example 7: let S be the set of all ordered pairs of real numbers with addition defined by

( x[1], x[2] ) + ( y[1], y[2] ) = ( x[1]+y[1], x[2]+y[2] ) and scalar multiplication r( x[1], x[2] ) = ( r*x[1], x[2] ).

Is S with these operations a Vector Space?

Must show closure and properties A1-A4 and S1-S4.

A1: (u + v) + w = u + (v + w)

A2: v + w = w + v

A3: 0 + v = v

A4: v + (-v) = 0

S1: r(v + w) = rv + rw

S2: (r + s)v = rv + sv

S3: r(sv) = (rs)v

S4: 1v = v

Example 8: Let S denote the set of positive real numbers. Define the operation of scalar

multiplication as,

(rx) = x^r and operation of addition as,

(x + y) = xy.

Is S a vector space with these operations?

Must show closure and properties A1-A4 and S1-S4.

A1: (u + v) + w = u + (v + w)

A2: v + w = w + v

A3: 0 + v = v

A4: v + (-v) = 0

S1: r(v + w) = rv + rw

S2: (r + s)v = rv + sv

S3: r(sv) = (rs)v

S4: 1v = v

Properties of Vector Spaces

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Theorem 3.1:

Every vector space V has the following properties:

1.) The vector 0 in V is unique.

2.) The vector -v in V is unique.

3.) If u + v = u + w then v = w.

4.) 0v = 0 for scalar 0 and vector v.

5.) r0 = 0 for scalar r and vector 0.

6.) (-r)v = r(-v) = -(rv) for scalar r and vector v.

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Exercises

1, 5, 9, 11, 13, 16, 18.