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Exercise: Let $C = (e_1, e_2, e_3)$ be the canonical basis of $\mathbb{R}^3$, and let $f$ be the endomorphism of $\mathbb{R}^3$ defined by: $f(x, y, z) = (2x-y, x+y+z, -x+2y+2z).$

1. Determine the matrix associated with $f$ with respect to the basis $C$.

2. Let $B = \{\varepsilon_1 = (0, 1, 1), \varepsilon_2 = (1, 0, 1), \varepsilon_3 = (1, 1, 0)\}$: i. Verify that $B$ is a basis of $\mathbb{R}^3$ and find the matrix associated with $f$ in the basis $B$.
   ii. Determine the matrix associated with $f$ with respect to the bases $B$ and $C$, and then the matrix associated with $f$ with respect to the bases $C$ and $B$.
   iii. Give the transition matrices from $B$ to $C$ and from $C$ to $B$.
   iv. Deduce the matrix associated with $f$ in the basis $C$.

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5 relative questions:

1. What is the general procedure for determining the matrix representation of a linear transformation with respect to a given basis?
2. How can you verify that a set of vectors forms a basis in $\mathbb{R}^n$?
3. What is the role of transition matrices between two bases?
4. How do you compute the matrix representation of a linear map in a new basis given the original matrix?
5. How is the canonical basis of $\mathbb{R}^n$ used as a reference in matrix transformations?

Tip: To verify a set is a basis, check that its vectors are linearly independent and span the space.