Question

v is a set of ordered pairs (a, b) of real numbers. Sum and scalar multiplication are defined by: (a, b) + (c, d) = (a + c, b + d) k (a, b) = (kb, ka)

Vectors
ANSWERED
asked 2021-06-20

v is a set of ordered pairs (a, b) of real numbers. Sum and scalar multiplication are defined by: \((a, b) + (c, d) = (a + c, b + d) k (a, b) = (kb, ka)\) (attention in this part) show that V is not linear space.

Answers (1)

2021-06-21

If V is a linear space, we must have that \(lv=v\),
for all v from V. Take for example \(v=(0,1)\)
Then \(lv=1(0,1)=(1 \cdot 1,1 \cdot 0)=(1,0)= v.\)
Thus V is not linear space.

0
 
Best answer

expert advice

Have a similar question?
We can deal with it in 3 hours

Relevant Questions

asked 2021-02-25

Let U and W be vector spaces over a field K. Let V be the set of ordered pairs (u,w) where \(u \in U\) and \(w \in W\). Show that V is a vector space over K with addition in V and scalar multiplication on V defined by
\((u,w)+(u',w')=(u+u',w+w')\ and\ k(u,w)=(ku,kw)\)
(This space V is called the external direct product of U and W.)

asked 2021-08-02
Express the following in set-builder notation:
a)The set A of natural numbers divisible by 3.
b)The set B of pairs (a,b) of real numbers such that a + b is an integer.
c)The open interval C = (—2,2).
d)The set D of 20 element subsets of N.
asked 2021-08-05
Express the following in set-builder notation in discrete math:
a)The set A of natural numbers divisible by 3.
b)The set B of pairs (a,b) of real numbers such that a + b is an integer.
c)The open interval C = (—2,2).
d)The set D of 20 element subsets of N.
asked 2021-06-10
Determine whether the given set S is a subspace of the vector space V.
A. V=\(P_5\), and S is the subset of \(P_5\) consisting of those polynomials satisfying p(1)>p(0).
B. \(V=R_3\), and S is the set of vectors \((x_1,x_2,x_3)\) in V satisfying \(x_1-6x_2+x_3=5\).
C. \(V=R^n\), and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.
D. V=\(C^2(I)\), and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.
E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.
F. V=\(P_n\), and S is the subset of \(P_n\) consisting of those polynomials satisfying p(0)=0.
G. \(V=M_n(R)\), and S is the subset of all symmetric matrices
...