Answered

Let u, v, and w be distinct vectors of a vector space V. Prove that if {u, v, w} is a basis for V, then {u + v, u + w, v + w} is also a basis for V. (Note: V is an arbitrary vector space, so you may not assume that u, v, and w are tuples.)

Answer :

LammettHash

[tex]\{u,v,w\}[/tex] forms a basis for [tex]V[/tex], which means any vector [tex]x\in V[/tex] can be written as the linear combination,

[tex]x=c_1u+c_2v+c_3w[/tex]

Pick [tex]c_1=d_1+d_2[/tex], [tex]c_2=d_1+d_3[/tex], and [tex]c_3=d_2+d_3[/tex]; then

[tex]x=(d_1+d_2)u+(d_1+d_3)v+(d_2+d_3)w[/tex]

[tex]x=d_1(u+v)+d_2(u+w)+d_3(v+w)[/tex]

Any [tex]x\in V[/tex] can thus be written as a linear combination of the vectors [tex]\{u+v,u+w,v+w\}[/tex], so these three vectors also form a basis for [tex]V[/tex].

ellakelly200

Answer:

vvv

Step-by-step explanation:

Other Questions