If i have a vector $u$ with coordinates $(2,3,4)$ and base $e_1$, $e_2$ and $e_3$, how can i decide the coordinates for the vector $u$ with base $e_1'$, $e_2'$, $e_3'$ if $$ e_2' = e_3, \quad e_3' = e_1 + e_3, \quad e_1' = e_1 + e_2 + e_3 $$ I'd appreciate any help as I've been stuck on this for a while.
Calculating vector coordinates with new base
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linear-algebra
vectors
1 Answers
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The fact that you know the coordinates of $u$ in the unprimed basis implies that $$u= 2e_1+3e_2+4e_3$$ You have expressions for $e_1, e_2, e_3$ in terms of the primed basis so all you need to do is substitute in and rearrange to find the coefficients of the $u$ expressed in terms of the primed basis.