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.

Question

asked 2021-02-25

\((u,w)+(u',w')=(u+u',w+w')\ and\ k(u,w)=(ku,kw)\)

(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.

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.

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

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