?Let \(\boldsymbol{V}\) be the set of all ordered pairs of real numbers, and consider

Chapter 4, Problem 1

(choose chapter or problem)

Let \(\boldsymbol{V}\) be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operations on \(\mathbf{u}=\left(u_{1}, u_{2}\right)\) and \(\mathbf{v}=\left(v_{1}, v_{2}\right)\):

\(\mathbf{u}+\mathbf{v}=\left(u_{1}+v_{1}, u_{2}+v_{2}\right), \quad k \mathbf{u}=\left(0, k u_{2}\right)\)

                 

                             

a. Compute \(\mathbf{u}+\mathbf{v}\) and \(k \mathbf{u}\) for \(\mathbf{u}=(-1,2), \mathbf{v}=(3,4)\), and \(k=3\).

 

b. In words, explain why \(\boldsymbol{V}\) is closed under addition and scalar multiplication.

c. Since addition on \(\boldsymbol{V}\) is the standard addition operation on \(R^{2}\) , certain vector space axioms  hold for \(\boldsymbol{V}\) because they are known to hold for \(R^{2}\) . Which axioms are they?

d. Show that Axioms 7, 8, and 9 hold.

e. Show that Axiom 10 fails and hence that \(\boldsymbol{V}\) is not a vector space under the given operations.

Equation Transcription:

𝑽

𝐮

𝐯

𝐮+𝐯𝐮

𝐮+𝐯

𝐮

𝐮𝐯

𝑽

𝑽

𝑽

𝑽

Text Transcription:

V

u=(u_1,u_2)

v=(v_1,v_2)

u+v=(u_1+v_1,u_2+v_2),  ku=(0,ku_2)

u+v

ku

u=(-1,2),v=(3,4)

k=3

V

V

R^2

V

R^2

V