Page 172 - Matrix Analysis & Applied Linear Algebra

4.1 Spaces and Subspaces 167

Proof. To prove (4.1.1), demonstrate that the two closure properties (A1) and
(M1) hold for S = X +Y. To show (A1) is valid, observe that if u, v ∈S, then
u = x 1 + y 1 and v = x 2 + y 2 , where x 1 , x 2 ∈X and y 1 , y 2 ∈Y. Because
X and Y are closed with respect to addition, it follows that x 1 + x 2 ∈X
and y 1 + y 2 ∈Y, and therefore u + v =(x 1 + x 2 )+(y 1 + y 2 ) ∈S. To
verify (M1), observe that X and Y are both closed with respect to scalar
multiplication so that αx 1 ∈X and αy 1 ∈Y for all α, and consequently
αu = αx 1 + αy 1 ∈S for all α. To prove (4.1.2), suppose S X = {x 1 , x 2 ,..., x r }
and S Y = {y 1 , y 2 ,..., y t } , and write

r t

z ∈ span (S X ∪S Y ) ⇐⇒z = α i x i + β i y i = x + y with x ∈X, y ∈Y
i=1 i=1
⇐⇒z ∈X + Y.

Example 4.1.8
2 2
If X⊆ and Y⊆ are subspaces defined by two different lines through
2
the origin, then X + Y = . This follows from the parallelogram law—sketch
a picture for yourself.
Exercises for section 4.1

n
4.1.1. Determine which of the following subsets of are in fact subspaces of
n
(n> 2).
(a) {x | x i ≥ 0}, (b) {x | x 1 =0}, (c) {x | x 1 x 2 =0},

n n

(d) x x j =0 , (e) x x j =1 ,
j=1 j=1
(f) {x | Ax = b, where A m×n = 0 and b m×1 = 0} .

n×n
4.1.2. Determine which of the following subsets of are in fact subspaces
n×n
of .
(a) The symmetric matrices. (b) The diagonal matrices.
(c) The nonsingular matrices. (d) The singular matrices.
(e) The triangular matrices. (f) The upper-triangular matrices.
(g) All matrices that commute with a given matrix A.
2
(h) All matrices such that A = A.
(i) All matrices such that trace (A)=0.
3
4.1.3. If X is a plane passing through the origin in and Y is the line
through the origin that is perpendicular to X, what is X + Y ?