1. coordinate vector help

Let $\displaystyle T:\mathbb{R^{3}}$$\displaystyle \rightarrow$$\displaystyle \mathbb{R^{3}}$ be the linear transformation defined by $\displaystyle T(x,y,z)=(2x+y,x+2z,x+y+z)$.

Q: Let $\displaystyle \beta=\{(1,0,1),(1,1,0),(0,1,1)\}$. Find the coordinate vectors of $\displaystyle e_{1},e_{2},e_{3}$ relative to $\displaystyle \beta$

Not sure how to put everything together, some guidance would be greatly appreciated.

2. Originally Posted by LexyLeia
Let $\displaystyle T:\mathbb{R^3}$$\displaystyle \rightarrow$$\displaystyle \mathbb{R^3}$ be the linear transformation defined by $\displaystyle T(x,y,z)=(2x+y,x+2z,x+y+z)$.

Q: Let $\displaystyle \beta=\{(1,0,1),(1,1,0),(0,1,1)\}$. Find the coordinate vectors of $\displaystyle e_{1},e_{2},e_{3}$ relative to $\displaystyle \beta$

Not sure how to put everything together, some guidance would be greatly appreciated.
The "coordinate vectors" of any vector u is the ordered triple (a, b, c) where a(1,0,1)+ b(1,1,0)+ c(0,1,1)= v.

So you need (a,b,c) such that a(1,0,1)+ b(1,1,0)= c(0,1,1)= (1, 0, 0).
That is the same as saying (a, 0, a)+ (b, b, 0)+ (0, c, c)= (a+b, b+ c, a+ c)= (1, 0, 0) or a+b= 1, b+ c= 0, a+ c= 0. Solve those equations for a, b, and c.

"T" has nothing to do with this question!

3. Originally Posted by HallsofIvy
So you need (a,b,c) such that a(1,0,1)+ b(1,1,0)= c(0,1,1)= (1, 0, 0).
Thanks for the response. How did you get the vector v=(1,0,0)?

4. You said "find the coordinate vector of $\displaystyle e_1$, $\displaystyle e_2$, and $\displaystyle e_3$" and those are usual notation for the standard basis vectors of $\displaystyle R^3$, (1, 0, 0), (0, 1, 0), and (0, 0, 1), respectively.