Proving: span{X_{1},...,X{N}} = span{Y,X_{2},...,X_{N}}

[stunner5000pt]

Suppose X_{1},X_{2},...,X_{N} ae vectors in Rn. If Y = a_{1} X_{1} ... + a_{N} X_{N} where ai is not zero, show that
span{X_{1},...,X{N}} = span{Y,X_{2},...,X_{N}}

WELL
span{X_{1},...,X{N}} = a_{1} X_{1} + ... + a_{N} X_{N}
Y = a_{1} X_{1} + ... + a_{N} X_{N}
then bX_{1} = Y - a_{2} X_{2} ... - a_{N} X_{N}

so i can see that bX_{1} = span{Y,X_{2},...,X_{N}}
also we know that X 1 has a unique representation as a span of the Xi, where i is not 1

but i m not sure how connect the two...

 

[0rthodontist]

Your notation is wrong. Span x1, ... xn is not equal to a1x1 ... anxn. Instead, you know that if v is an ELEMENT of Span x1, ... xn, then v can be written as a1x1 ... anxn with not all of the ai's zero.

You are trying to show that a vector v is in Span x1, ... xn, if and only if it is in span y, x2, ... xn.