Span of two vectors is the same as the Span of the linear combination of those two vectors. Question: Let vec(u) ,vec(v) in RR^n. Span{vec(u) ,vec(v)}=Span{vec(u) +vec(v) , vec(u) −vec(v)}

skilpadw3

skilpadw3

Answered question

2022-07-18

Question: Let u , v R n . S p a n { u , v } = S p a n { u + v , u v }
I tried to approach this proof by finding a linear combination of u and v but I am confused as to how to approach the linear combination of the right hand side of the equation. Please help!

Answer & Explanation

Ali Harper

Ali Harper

Beginner2022-07-19Added 16 answers

Well, the RHS is quite trivially a subset of the LHS. Conversely, you can recover both x and y very easily from the vectors x + y and x y. Namely, 1 / 2 ( x + y ) + 1 / 2 ( x y ) = x. Then once you have x, you can subtract it from x + y to get y. Thus we have the reverse inclusion.
Hence they are equal.

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