The Inner Product on V ⊗ W: Multiply the Factors

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1. Question 1

   In $\mathbf{R}^3 \otimes \mathbf{R}^2$ (standard inner products on the factors), let $v = (1, 2, 0)$, $u = (0, 1, 3)$, $w = (2, -1)$, $x = (1, 1)$. Find $\langle v \otimes w,\ u \otimes x \rangle$.

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2. Question 2

   In $\mathbf{R}^3 \otimes \mathbf{R}^2$, let $v = (2, 1, 2)$ and $w = (3, 4)$. Find $\|v \otimes w\|$.

   $\|v \otimes w\| =$

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3. Question 3

   In $\mathbf{C}^2 \otimes \mathbf{C}^2$ (standard inner products on the factors), find $\|(1, i) \otimes (2, 1 + i)\|^2$.

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4. Question 4

   In $\mathbf{R}^2 \otimes \mathbf{R}^2$ with the standard inner products, identify each element with its coefficient grid in the basis $e_j \otimes f_k$. Find $\langle A, B \rangle$ for $A = \begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 0 \\ 1 & 2 \end{pmatrix}$.

   $\langle A, B \rangle =$

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5. Question 5

   Let $e_1 = \tfrac{1}{5}(3, 4)$, $e_2 = \tfrac{1}{5}(4, -3)$ (an orthonormal basis of $\mathbf{R}^2$) and let $f_1, f_2$ be the standard basis of $\mathbf{R}^2$. Find $\|2\, e_1 \otimes f_1 - e_2 \otimes f_2 + 2\, e_2 \otimes f_1\|$.

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