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Coordinates in a basis

Solve the first problem 6 problems

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1. Dot product using a dual basis ●○○○○ 10 ptsLet $B=\{\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3,\mathbf{e}_4\}$ be a basis of $\mathbb{R}^4$ with $\mathbf{e}_1=(1,0,0,0)$, $\mathbf{e}_2=(1,1,0,0)$, $\mathbf{e}_3=(1,1,1,0)$, $\mathbf{e}_4=(1,1,1,1)$. Its dual basis is $\mathbf{e}^1=(1,-1,0,0)$, $\mathbf{e}^2=(0,1,-1,0)$, $\mathbf{e}^3=(0,0,1,-1)$, $\mathbf{e}^4=(0,0,0,1)$. Let $\mathbf{u}=\mathbf{e}^1-\mathbf{e}^2+\mathbf{e}^3-\mathbf{e}^4$ and $\mathbf{v}=(1,2,3,4)$, where vectors are written in the standard basis of $\mathbb{R}^4$. Find $\mathbf{u}\cdot\mathbf{v}$ using the standard dot product.
2. 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 $v$.
3. Coefficients of a vector in a basis ●●○○○ 20 ptsGiven vectors $\vec{a} = (1, -1)$, $\vec{b} = (2, 1)$ and $\vec{c} = (-2, 1)$, suppose $\vec{c} = x\vec{a} + y\vec{b}$ with $x, y \in \mathbb{R}$. Find $x - y$.
4. 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 $x=\alpha p+\beta q+\gamma r$.
5. 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 $v = \alpha_1 a_1 + \alpha_2 a_2 + \alpha_3 a_3$.

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