Defining V ⊗ W as Bilinear Functionals on V′ × W′

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1. Question 1

   In $\mathbf{F}^3 \otimes \mathbf{F}^2$, let $v = (1, 0, 2)$, $w = (3, -1)$, $\varphi(x) = 2x_1 + x_3$ and $\tau(y) = y_1 + 2y_2$. Find $(v \otimes w)(\varphi, \tau)$.

   $(v \otimes w)(\varphi, \tau) =$

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2. Question 2

   Let $V = \mathcal{L}(\mathbf{F}^2, \mathbf{F}^3)$ and $W = \mathcal{P}_1(\mathbf{F})$. Find $\dim(V \otimes W)$, where $V \otimes W = \mathcal{B}(V', W')$.

   $\dim(V \otimes W) =$

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3. Question 3

   Suppose $(v \otimes w)(\varphi, \tau) = 7$. Find $\big((3v) \otimes (-2w) + v \otimes (4w)\big)(\varphi, \tau)$.

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4. Question 4

   For $v_1, v_2 \in V$ and $w_1, w_2 \in W$, which element of $V \otimes W$ equals $(v_1 + v_2) \otimes (w_1 - w_2)$?

   a

   $v_1 \otimes w_1 - v_2 \otimes w_2$

   b

   $v_1 \otimes w_1 - v_1 \otimes w_2 + v_2 \otimes w_1 + v_2 \otimes w_2$

   c

   $v_1 \otimes w_1 - v_1 \otimes w_2 + v_2 \otimes w_1 - v_2 \otimes w_2$

   d

   $v_1 \otimes w_1 - v_1 \otimes w_2 - v_2 \otimes w_1 + v_2 \otimes w_2$

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5. Question 5

   In $\mathbf{F}^2 \otimes \mathbf{F}^2$, let $t = (1, 2) \otimes (0, 1) + (3, -1) \otimes (2, 2)$, $\varphi(x) = x_1 + x_2$ and $\tau(y) = 2y_1 + y_2$. Find $t(\varphi, \tau)$.

   $t(\varphi, \tau) =$

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