A Basis of V ⊗ W, and v ⊗ w as a Grid of Products

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1. Question 1

   In $\mathbf{F}^2 \otimes \mathbf{F}^3$ with the standard bases $e_1, e_2$ and $f_1, f_2, f_3$, let $$t = (3, -2) \otimes (1, 4, 5) + (1, 1) \otimes (2, 0, -3).$$ Find the coefficient of $e_2 \otimes f_3$ in $t$.

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2. Question 2

   Let $e_1, e_2$ and $f_1, f_2$ be bases of $V$ and $W$ with dual bases $\varphi_1, \varphi_2$ and $\tau_1, \tau_2$. Let $t = 2\, e_1 \otimes f_1 - e_1 \otimes f_2 + 5\, e_2 \otimes f_1 + 3\, e_2 \otimes f_2$. Find $t(\varphi_1 + 2\varphi_2,\ \tau_1 - \tau_2)$.

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3. Question 3

   Each matrix is the coefficient grid of an element of $\mathbf{F}^2 \otimes \mathbf{F}^2$ in the basis $e_j \otimes f_k$. Which one is a simple tensor $v \otimes w$?

   a

   $\begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$

   b

   $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

   c

   $\begin{pmatrix} 2 & 4 \\ 3 & 5 \end{pmatrix}$

   d

   $\begin{pmatrix} 2 & 3 \\ 4 & 6 \end{pmatrix}$

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4. Question 4

   The coefficient grid of $u \in \mathbf{F}^2 \otimes \mathbf{F}^3$ in the standard basis $e_j \otimes f_k$ is $\begin{pmatrix} 3 & 6 & -3 \\ -1 & -2 & 1 \end{pmatrix}$. Which simple tensor is $u$?

   a

   $(3, -1) \otimes (1, 2, -1)$

   b

   $(1, 2, -1) \otimes (3, -1)$

   c

   $(3, 1) \otimes (1, 2, -1)$

   d

   $(1, -1) \otimes (3, 6, -3)$

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5. Question 5

   For which number $c$ is the element with coefficient grid $\begin{pmatrix} 2 & c \\ 3 & 6 \end{pmatrix}$ (in the basis $e_j \otimes f_k$ of $\mathbf{F}^2 \otimes \mathbf{F}^2$) a simple tensor?

   $c =$

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