Let U and V be the subspaces of R n a ) Show that the union, U ∪ V , satisfies properties ( s 1 ) and ( s 3 ) of Theorem 2. b ) If neither U nor V is a subset of the other, show that U ∪ V does not satisfy condition ( s 2 ) of Theorem 2. Theorem 2: A subset W of R n if and only if the following conditions are met: ( s 1 ) The zero vector , θ , is in W . ( s 2 ) x + y is in W whenever x and y are in W . ( s 3 ) a x is in W whenever x is in W and a is any scalar.
Let U and V be the subspaces of R n a ) Show that the union, U ∪ V , satisfies properties ( s 1 ) and ( s 3 ) of Theorem 2. b ) If neither U nor V is a subset of the other, show that U ∪ V does not satisfy condition ( s 2 ) of Theorem 2. Theorem 2: A subset W of R n if and only if the following conditions are met: ( s 1 ) The zero vector , θ , is in W . ( s 2 ) x + y is in W whenever x and y are in W . ( s 3 ) a x is in W whenever x is in W and a is any scalar.
Solution Summary: The author proves that for two subspaces U and V of a vector space Rn, their union
a
)
Show that the union,
U
∪
V
, satisfies properties
(
s
1
)
and
(
s
3
)
of Theorem 2.
b
)
If neither
U
nor
V
is a subset of the other, show that
U
∪
V
does not satisfy condition
(
s
2
)
of Theorem 2.
Theorem 2: A subset
W
of
R
n
if and only if the following conditions are met:
(
s
1
)
The zero vector,
θ
, is in
W
.
(
s
2
)
x
+
y
is in
W
whenever
x
and
y
are in
W
.
(
s
3
)
a
x
is in
W
whenever
x
is in
W
and
a
is any scalar.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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