192 CHAPTER 7. VECTOR SPACES AND FIELDS

Proof 3: Suppose r > s. Let zk denote a vector of {y1, · · · ,ys} . Thus there exists j assmall as possible such that

span (y1, · · · ,ys) = span (x1, · · · ,xm, z1, · · · , zj)

where m + j = s. It is given that m = 0, corresponding to no vectors of {x1, · · · ,xm} andj = s, corresponding to all the yk results in the above equation holding. If j > 0 then m < sand so

xm+1 =

m∑k=1

akxk +

j∑i=1

bizi

Not all the bi can equal 0 and so you can solve for one of them in terms of xm+1,xm, · · · ,x1,and the other zk. Therefore, there exists

{z1, · · · , zj−1} ⊆ {y1, · · · ,ys}

such thatspan (y1, · · · ,ys) = span (x1, · · · ,xm+1, z1, · · · , zj−1)

contradicting the choice of j. Hence j = 0 and

span (y1, · · · ,ys) = span (x1, · · · ,xs)

It follows thatxs+1 ∈ span (x1, · · · ,xs)

contrary to the assumption the xk are linearly independent. Therefore, r ≤ s as claimed. ■

Corollary 7.2.5 If {u1, · · · ,um} and {v1, · · · ,vn} are two bases for V, then m = n.

Proof: By Theorem 7.2.4, m ≤ n and n ≤ m. ■

Definition 7.2.6 A vector space V is of dimension n if it has a basis consisting of n vectors.This is well defined thanks to Corollary 7.2.5. It is always assumed here that n <∞ and inthis case, such a vector space is said to be finite dimensional.

Example 7.2.7 Consider the polynomials defined on R of degree no more than 3, denotedhere as P3. Then show that a basis for P3 is

{1, x, x2, x3

}. Here xk symbolizes the function

x 7→ xk.

It is obvious that the span of the given vectors yields P3. Why is this set of vectorslinearly independent? Suppose

c0 + c1x+ c2x2 + c3x

3 = 0

where 0 is the zero function which maps everything to 0. Then you could differentiate threetimes and obtain the following equations

c1 + 2c2x+ 3c3x2 = 0

2c2 + 6c3x = 0

6c3 = 0

Now this implies c3 = 0. Then from the equations above the bottom one, you find insuccession that c2 = 0, c1 = 0, c0 = 0.

There is a somewhat interesting theorem about linear independence of smooth functions(those having plenty of derivatives) which I will show now. It is often used in differentialequations.