I have this exercise.
Consider the following vectors of ℝ²:
$$ u_1 = \begin{pmatrix}1\\2\end{pmatrix}, \space\space\space u_2 = \begin{pmatrix}2\\1\end{pmatrix} $$
Determine the coordinates of the vector v relative to the basis B.
$$ v = \begin{pmatrix}1\\1\end{pmatrix}, \space\space\space B = \begin{pmatrix}u_1, \space u_2\end{pmatrix} $$
Which of the two is the correct resolution?
Resolution 1: $$ v_B = v_1 u_1 + v_2 u_2 \\ v_B = 1 \begin{pmatrix}1\\2\end{pmatrix} + 1 \begin{pmatrix}2\\1\end{pmatrix} \\ v_B = \begin{pmatrix}3\\3\end{pmatrix} $$
Resolution 2:
$$ v_B = α u_1 + β u_2, \space\space α,β∈ℝ \\ \begin{cases} α + 2β = 1 \\ 2α + β = 1 \end{cases} \\ α = β = \frac{1}{3} \\ v_B = \begin{pmatrix}\frac{1}{3}\\\frac{1}{3}\end{pmatrix} $$
UPDATE: with the responses I understand that the right solution is the second, but I still have a doubt. I have found an exercise on my book, similar to the previous, but the solution is like the first:
Consider the following vectors:
$$ v_1 = \begin{pmatrix}1\\0\\-1\end{pmatrix}, \space\space\space v_2 = \begin{pmatrix}0\\1\\-1\end{pmatrix} $$
Determine the coordinates of the vector w ∈ $_O^3$ relative to the basis B.
$$ v = \begin{pmatrix}1\\2\end{pmatrix}, \space\space\space B = \begin{pmatrix}v_1, \space v_2\end{pmatrix} $$
The resolution by the book is the following:
$$ w_B = 1v_1 + 2v_2 = (\hat{ı}-\hat{k}) + 2(\hat{ȷ}-\hat{k}) = \hat{ı} + 2\hat{ȷ} - 3\hat{k} $$
that seems the previous Resolution 1:
$$ w_B = w_1 v_1 + w_2 v_2 \\ w_B = 1 \begin{pmatrix}1\\0\\-1\end{pmatrix} + 2 \begin{pmatrix}0\\1\\-1\end{pmatrix} \\ w_B = \begin{pmatrix}1\\2\\-3\end{pmatrix} $$
At some point one should have $$ v = v_1 \, u_1 + v_2 \, u_2 $$ where the $v_i$ are components of $v$, and the $u_i$ are base vectors.
The first option is a nice distraction, because $v$ is given in coordinates of the standard base $e_i$ and not in coordinates of the base $B$.
So indeed the second one is true, as can be tried out by $$ v = v_1^B \, u_1 + v_2^B \, u_2 $$ with the coordinates $v_i^B = 1/3$ regarding $B$.
The difficulty here is not the maths, but the lack of precision of the problem specification, here: regarding to what base the given coordinate tuples are specified.
The base vectors are given in coordinates of the canonical base, if they were given regarding to another base, and the only other base in question is $B$, their coordinates would have been $(u_i)_j =\delta_{ij}$ which they are not.
For $v$ we can not tell which base the coordinates refer to, there is no hint in the text or the notation. By convention we expect that we stick to the same coordinate system like the other given vectors, thus standard base.
For the problem from the update the situation is a bit different. That $v$ or rather $w$ has two coordinates while the base vectors have three. Further there are only two base vectors. This makes only sense for coordinates of a plane. So that vector is given to that $B$.