Let x,y and z be three linearly independent vectors. Explain, with justification, whether or not span ⁡ { x , y , z } = span ⁡ { x + y , y + z , z + x }

Question

Let x,y and z be three linearly independent vectors. Explain, with justification, whether or not   span  ⁡  {  x  ,    y  ,    z  }  =  span  ⁡  {  x  +  y  ,    y  +  z  ,    z  +  x  }

Royce Morrison · Accepted Answer

x  =      1    2    (  x  +  y  )  −      1    2    (  y  +  z  )  +      1    2    (  z  +  x  )  y  =      1    2    (  x  +  y  )  −      1    2    (  z  +  x  )  +      1    2    (  y  +  z  )  z  =      1    2    (  y  +  z  )  −      1    2    (  x  +  y  )  +      1    2    (  z  +  x  )So x,y,z can be written as a linear combination of (x+y),(z+x),(y+z) , hence the spans are equal.

Silvina2b · Accepted Answer

Assume   W  :=  v  ∈  s  p  a  n  {  x  +  y  ,  y  +  z  ,  z  +  x  } thenv = a(x+y) + b(y+z) + c(z+x)= (a+c)x + (a+b)y + (b+c)zso that if   v  ∈  W then   v  ∈  U, i.e.   W  ⊆  U. Assume   v  ∈  U thenv = ax + by + cz  =  (  (  a  +  b  −  c  )      /    2  )  (  x  +  y  )  +  (  (  b  +  c  −  a  )      /    2  )  (  y  +  z  )  +  (  (  a  −  b  +  c  )      /    2  )  (  z  +  x  )  ∈  Wi.e.   U  ⊆  W

Let x,y and z be three linearly independent vectors. Explain, with justification, whether or not span{x,y,z}=span{x+y,y+z,z+x}.

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