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1. Coordinates of a vector in R^4 ●○○○○ 10 ptsLet $a_1 = [1, 1, 0, 0]$, $a_2 = [0, 1, 1, 0]$ and $a_3 = [0, 0, 1, 1]$. Find the coefficients $\alpha_1, \alpha_2, \alpha_3$ such that $v = [1, 3, 5, 3] = \alpha_1 a_1 + \alpha_2 a_2 + \alpha_3 a_3$, that is, find the coordinate representation $[\alpha_1, \alpha_2, \alpha_3]$ of $v$.
2. Decomposing a vector in a basis ●●○○○ 20 ptsWrite the vector $x=(-5, -5, 5)$ as a linear combination of $p=(-2, 0, 1)$, $q=(1, 3, -1)$ and $r=(0, 4, 1)$, that is, find $\alpha$, $\beta$, $\gamma$ with $x=\alpha p+\beta q+\gamma r$.
3. Coordinates in a non-standard basis ●●○○○ 20 ptsLet $v = [6, 3, 2, 5]$, $a_1 = [2, 2, 2, 3]$, $a_2 = [1, 0, -1, 0]$ and $a_3 = [0, 1, 0, 1]$. Find the coordinates $[\alpha_1, \alpha_2, \alpha_3]$ of $v$ in terms of $a_1, a_2, a_3$, that is, the coefficients with $v = \alpha_1 a_1 + \alpha_2 a_2 + \alpha_3 a_3$.