Principles of Mathematics in Operations Research

2.1 Vector Spaces 15

ii. Take any w. Let e*i = -1, a 2 = 0, and wi = w. Then,

(-l)-w + (0)-w 2 =-w eW. D

Example 2.1.8 S C R2x3, consisting of the matrices of the form

0 P 7

a a - P a + 27

is a subspace of j>2x3

Proposition 2.1.9 IfWx,W 2 are subspaces, then so is W\ l~l W 2.

Proof. Take u>i, u> 2 € Wi n W 2 , ai, a 2 £ F.

i. wi, w 2 G Wi =>• ai • wi + a 2 • w 2 € Wi

ii. wi,w 2 e W 2 => cti • Wi + a 2 • w 2 £ W 2

Thus, aitui + a 2 w 2 € Wi n W 2. •

Remark 2.1.10 IfW\, W 2 are subspaces, then W\ UW 2 is not necessarily a

subspace.

Definition 2.1.11 Let V be a vector space over ¥, X C V. X is said to

be linearly dependent if there exists a distinct set of xi,x 2 ,... ,Xk £ X and

scalars a\,a 2 , ...,atk 6 F not all zero 3 5^i=1 o^Xi = 9. Otherwise, for any

subset of size k,

k

X\,X 2 ,...,Xk £ X, 2_2aixi — ® => al — a2 = ••• = <*k = 0.

In this case, X is said to be linearly independent.

We term an expression of the form $Zi=1 ot{Xi as linear combination.

In particular, if JZi=i ai — 1» we ca^ ^ affine combination. Moreover, if

Si=i ai = 1 and ai > 0, Vi = 1,2, ...,k, it becomes convex combination.

On the other hand, if a* > 0, Vi = 1,2,..., k; then X)=i

said to be

canonical combination.

Example 2.1.12 In Rn, let E = {e;}"=1 where ef = (0, • • • 0,1,0, • • • , 0) is

the ith canonical unit vector that contains 1 in its ith position and 0s elsewhere.

Then, E is an independent set since

aiei H ha„en =

«i

a„

at = 0, Vi

Let X = {xi}"=1 where xf = (0, • • -0,1,1, • • • , 1) is the vector that con-
tains 0s sequentially up to position i, and it contains Is starting from position
i onwards. X is also linearly independent since