- #1
stunner5000pt
- 1,461
- 2
Suppose [itex] X_{1},X_{2},...,X_{N} [/itex] ae vectors in Rn. If [itex] Y = a_{1} X_{1} ... + a_{N} X_{N} [/itex] where ai is not zero, show that
[tex] span{X_{1},...,X{N}} = span{Y,X_{2},...,X_{N}} [/tex]
WELL
[tex] span{X_{1},...,X{N}} = a_{1} X_{1} + ... + a_{N} X_{N} [/tex]
[tex] Y = a_{1} X_{1} + ... + a_{N} X_{N} [/tex]
then [tex] bX_{1} = Y - a_{2} X_{2} ... - a_{N} X_{N} [/tex]
so i can see that [tex] bX_{1} = span{Y,X_{2},...,X_{N}} [/tex]
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...
[tex] span{X_{1},...,X{N}} = span{Y,X_{2},...,X_{N}} [/tex]
WELL
[tex] span{X_{1},...,X{N}} = a_{1} X_{1} + ... + a_{N} X_{N} [/tex]
[tex] Y = a_{1} X_{1} + ... + a_{N} X_{N} [/tex]
then [tex] bX_{1} = Y - a_{2} X_{2} ... - a_{N} X_{N} [/tex]
so i can see that [tex] bX_{1} = span{Y,X_{2},...,X_{N}} [/tex]
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...