Question

Determine whether the set S is subspace of R5 defined by

S = f(x1; x2; x3; x4; x5) R5

: x1 = 3x2 and x3 = 7x4:

Answer 1

Let's take a vectors a = (3*a2, a2, 7*a4, a4, a5) and b = (3*b2, b2, 7*b4, b4, b5), $a, b \in S$

To be a subspace, next conditions suppose to be met:

• dim of vectors = 5

$a + b \in S$ obviously a+b

• a+b∈S 

a+b = (3(a2+b2), a2+b2, 7(a4+b4), a4+b4, a5+b5))

let xn = an+bn, then:

a+b = (3*x2, x2, 7*x4, x4, x5) obviously $a+b \in S$

• $s*a\in S, s - scalar$

s*a = (3*s*a2, s*a2, 7*s*a4, s*a4, s*a5)

let xn = s*an, then:

s*a = (3*x2, x2, 7*x4, x4, x5) obviously $s*a \in S$

S is a subspace of R5.