Show that span{X₁,...,Xₙ} = span{Y,X₂,...,Xₙ} when Y is a linear combination

[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.