Mathematics: 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.)

Let u, v, and w be distinct vectors of a vector

lindseyreneesmith7

19.01.2022

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

x=c_1u+c_2v+c_3w

Pick c_1=d_1+d_2 , c_2=d_1+d_3 , and c_3=d_2+d_3 ; then

x=(d_1+d_2)u+(d_1+d_3)v+(d_2+d_3)w

x=d_1(u+v)+d_2(u+w)+d_3(v+w)

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

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Learsyguerra26

18.03.2021

18/30 and 36/60 is an answer

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