Tensor Products of m Spaces and m-Linear Maps

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1. Question 1

   Let $V_1 = \mathbf{F}^3$, $V_2 = \mathcal{P}_2(\mathbf{F})$ (polynomials of degree at most $2$) and $V_3 = \mathbf{F}^{1,2}$ ($1$-by-$2$ matrices). Find $\dim(V_1 \otimes V_2 \otimes V_3)$.

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2. Question 2

   $\beta$ is a $3$-linear functional with $\beta(u, v, w) = 2$ and $\beta(u', v, w) = -1$. Find $\beta(3u - 2u',\ -v,\ 2w)$.

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3. Question 3

   Let $v_1 = (1, -1) \in \mathbf{F}^2$, $v_2 = (2, 0, 1) \in \mathbf{F}^3$, $v_3 = (3, 1) \in \mathbf{F}^2$, and $\varphi_1(x) = x_1 - x_2$, $\varphi_2(x) = x_1 + x_3$, $\varphi_3(x) = 2x_1 - x_2$. Find $(v_1 \otimes v_2 \otimes v_3)(\varphi_1, \varphi_2, \varphi_3)$.

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4. Question 4

   With standard bases $e^1_j$ of $\mathbf{F}^2$, $e^2_j$ of $\mathbf{F}^3$, $e^3_j$ of $\mathbf{F}^2$, find the coefficient of $e^1_2 \otimes e^2_3 \otimes e^3_1$ in $(1, 4) \otimes (2, -1, 3) \otimes (5, 2)$.

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5. Question 5

   A $3$-linear functional $\beta$ on $\mathbf{F}^2 \times \mathbf{F}^2 \times \mathbf{F}^2$ has $\beta(e_i, e_j, e_k) = 1$ when $i = j = k$ and $0$ otherwise ($e_1, e_2$ the standard basis). Find $\beta\big((1, 2), (3, 1), (2, -1)\big)$.

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