The Universal Property: Bilinear Maps Become Linear

Work each one out before you check it. Every question has a scratch pad and a calculator.

1. Question 1

   Let $\gamma \colon \mathbf{F}^2 \times \mathbf{F}^2 \to \mathbf{F}$, $\gamma(v, w) = v_1 w_1 + 2 v_2 w_1 - v_2 w_2$, with standard bases $e_1, e_2$ and $f_1, f_2$. Find $\hat{\gamma}(3\, e_1 \otimes f_1 - e_2 \otimes f_2 + 2\, e_2 \otimes f_1)$.

   $\hat{\gamma} =$

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2. Question 2

   A linear map $T \colon \mathbf{F}^2 \otimes \mathbf{F}^2 \to \mathbf{F}$ has values $T(e_1 \otimes f_1) = 1$, $T(e_1 \otimes f_2) = -2$, $T(e_2 \otimes f_1) = 0$, $T(e_2 \otimes f_2) = 3$ on the standard basis tensors. Find $T^\sharp\big((2, 1), (1, 1)\big)$.

   $T^\sharp =$

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3. Question 3

   On $\mathbf{F}^2$, let $S = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $T = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$ (matrices in the standard basis). Let $\Phi \colon \mathcal{L}(\mathbf{F}^2) \otimes \mathcal{L}(\mathbf{F}^2) \to \mathcal{L}(\mathbf{F}^2)$ be the linear map with $\Phi(A \otimes B) = AB$. Find the entry in row $1$, column $2$ of the matrix of $\Phi(S \otimes T + T \otimes S)$.

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4. Question 4

   Let $\Phi \colon \mathbf{F}^2 \otimes \mathbf{F}^{2,2} \to \mathbf{F}^2$ be the linear map with $\Phi(v \otimes A) = Av$. For $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ -1 & 2 \end{pmatrix}$, find $\Phi(e_1 \otimes A + e_2 \otimes B)$, where $e_1, e_2$ is the standard basis of $\mathbf{F}^2$.

   first entry

   second entry

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5. Question 5

   Let $\gamma(v, w) = v_1 w_1 + v_2 w_2 + v_1 w_2$ on $\mathbf{F}^2 \times \mathbf{F}^2$ and let $t = (1, 2) \otimes (3, 0) + (0, 1) \otimes (1, 1)$. Find $\hat{\gamma}(t)$.

   $\hat{\gamma}(t) =$

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