website: www.studysmarter.uwa.edu.au → Numeracy → Online Resources
Subspaces Solutions
These exercises have been written to consolidate your understanding of the Subspaces
workshop.
Question 1
For each of the following sets, try to guess whether it represents a subspace. Then prove
that it is or is not a subspace.
(a)
{(x, y, z) ∈ R3 : x = 4y + z}
(1) 1st entry in u + v
(2) 1st entry in αu
(b)
=
=
=
=
{(x, y, z) ∈ R3 : z = 0}
(1) 3rd entry in u + v
(c)
{(x, y, ) ∈ R2 : x4 = y 4 }
Subspace
= u1 + v1
= 4u2 + u3 + 4v2 + v3
= 4(u2 + v2 ) + (u3 + v3 )
= 4(2nd entry in u + v) + (3rd entry in u + v) X
αu1
α(4u2 + u3 )
4(αu2 ) + (αu3 )
4(2nd entry in αu) + (3rd entry in αu) X
Subspace
= 0+0
= 0X
(2) 3rd entry in αu = α × 0
= 0X
Not a subspace
For example (1, −1) + (1, 1) = (2, 0), which is not in the set.
Hence, the set is not closed under addition.
(d) {(x, y, z) ∈ R3 : x + y + z ∈ Z}
Not a subspace
For example 0.5 × (1, 0, 0) = (0.5, 0, 0). Since 0.5 + 0 + 0 = 0.5, which is not
an integer, the set is not closed under scalar multiplication.
(e)
{(x, y, z) ∈ R3 : 4 + zy = x}
Not a subspace
For example, the vectors (4, 1, 0) and (4, 0, 1) are in the set but their sum is not.
(f)
{(x, y, z) ∈ R3 : x =
y+z
}
2
(1) 1st entry in u + v
Subspace
= u1 + v1
3
3
= u2 +u
+ v2 +v
2
2
3 +v3 )
= (u2 +v2 )+(u
2
(2nd entry in u + v)+(3rd entry in u + v)
(2) 1st entry in αu
(g)
=
X
2
= αu1
3
= α u2 +u
2
(αu2 )+(αu3 )
=
2
(2nd entry in αu)+(3rd entry in αu)
X
=
2
{(x, y, z) ∈ R3 : x3 − y 3 = 0}
Subspace!
The condition reduces to x = y which can easily be shown to be a subspace.
(h) {(x, y, ) ∈ R2 : x2 + 2xy + y 2 = 0}
Subspace!
At first glance, the presence of squared terms and products would suggest
that this is not a subspace but the condition is the perfect square
(x + y)2 = 0 which reduces to x + y = 0 which can easily be shown to be a
subspace.
(i)
{(w, x, y, z) ∈ R4 : xy + z = 0}
Not a subspace
For example, the vectors (0, 1, 0, 0) and (0, 0, 1, 0) are in the set but their sum
is not.
Using STUDYSmarter Resources
This resource was developed for UWA students by the STUDYSmarter team for the
numeracy program. When using our resources, please retain them in their original form
with both the STUDYSmarter heading and the UWA crest.