Why V × W Fails: Bilinear Functionals and dim B(V, W)

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1. Question 1

   Let $\beta(v, w) = v_1 w_2 + 2 v_2 w_1$ on $\mathbf{R}^2 \times \mathbf{R}^2$, and take the two elements $\big((1, 1), (2, 0)\big)$ and $\big((0, 1), (1, 3)\big)$ of $\mathbf{R}^2 \times \mathbf{R}^2$. Add them in $\mathbf{R}^2 \times \mathbf{R}^2$ and apply $\beta$ to the sum; then subtract the sum of $\beta$ at the two elements. Find this difference.

   difference $=$

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2. Question 2

   Let $V = \mathcal{P}_2(\mathbf{F})$ (polynomials of degree at most $2$) and $W = \mathbf{F}^{2,3}$ ($2$-by-$3$ matrices). Find $\dim \mathcal{B}(V, W)$, the dimension of the space of bilinear functionals on $V \times W$.

   $\dim \mathcal{B}(V, W) =$

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3. Question 3

   Which function $\beta \colon \mathbf{R}^2 \times \mathbf{R}^2 \to \mathbf{R}$, with $v = (v_1, v_2)$ and $w = (w_1, w_2)$, is a bilinear functional?

   a

   $\beta(v, w) = (v_1 + w_1)(v_2 + w_2)$

   b

   $\beta(v, w) = 4 v_2 w_1 - v_1 w_2$

   c

   $\beta(v, w) = v_1 w_1 + v_2$

   d

   $\beta(v, w) = v_1 w_1 w_2$

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4. Question 4

   Let $\beta(v, w) = 3 v_1 w_2 - v_2 w_1 + 4 v_2 w_2$ on $\mathbf{R}^2 \times \mathbf{R}^2$. Use the basis $e_1 = (1, 1)$, $e_2 = (1, -1)$ of the first $\mathbf{R}^2$ and the standard basis $f_1, f_2$ of the second. Find the entry of $\mathcal{M}(\beta)$ in row $2$, column $1$.

   $\mathcal{M}(\beta)_{2,1} =$

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5. Question 5

   With the standard bases of $\mathbf{F}^3$ and $\mathbf{F}^2$, let $C = \begin{pmatrix} 2 & -1 \\ 0 & 3 \\ 1 & 1 \end{pmatrix}$ and $\beta_C(a, b) = \sum_{j=1}^{3} \sum_{k=1}^{2} C_{j,k}\, a_j b_k$. Find $\beta_C\big((1, 2, -1),\ (2, 1)\big)$.

   $\beta_C =$

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