Let R ∞ be the set of all infinite sequences of real numbers, with the operations u + v = ( u 1 , u 2 , u 3 , ...... ) + ( v 1 , v 2 , v 3 , ...... ) = ( u 1 + v 1 , u 2 + v 2 , u 3 + v 3 , ..... ) and c u = c ( u 1 , u 2 , u 3 , ...... ) = ( c u 1 , c u 2 , c u 3 , ...... ) . Determine whether R ∞ is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
Let R ∞ be the set of all infinite sequences of real numbers, with the operations u + v = ( u 1 , u 2 , u 3 , ...... ) + ( v 1 , v 2 , v 3 , ...... ) = ( u 1 + v 1 , u 2 + v 2 , u 3 + v 3 , ..... ) and c u = c ( u 1 , u 2 , u 3 , ...... ) = ( c u 1 , c u 2 , c u 3 , ...... ) . Determine whether R ∞ is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
Solution Summary: The author explains that the given set is a vector space. If the listed axioms are satisfied for every u,vandw in V and every scalar (
Let
R
∞
be the set of all infinite sequences of real numbers, with the operations
u
+
v
=
(
u
1
,
u
2
,
u
3
,
......
)
+
(
v
1
,
v
2
,
v
3
,
......
)
=
(
u
1
+
v
1
,
u
2
+
v
2
,
u
3
+
v
3
,
.....
)
and
c
u
=
c
(
u
1
,
u
2
,
u
3
,
......
)
=
(
c
u
1
,
c
u
2
,
c
u
3
,
......
)
.
Determine whether
R
∞
is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
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.
Suppose you flip a fair two-sided coin four times and record the result.
a). List the sample space of this experiment. That is, list all possible outcomes that could
occur when flipping a fair two-sided coin four total times. Assume the two sides of the coin are
Heads (H) and Tails (T).
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