(Solved): Recall that the direct sum V1V2 is defined to be the space of pairs ...

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Recall that the direct sum $$V_1?V_2$$ is defined to be the space of pairs of vectors $$\left(v_1,v_2\right)$$ such that $$v_1?V_1,v_2?V_2$$ where the vector space structure is defined by the componentwise addition and simultaneous scalar multiplication. (1) Show that there is an isomorphism $$U?\left(V_1?V_2\right)?\left(U?V_1\right)?\left(U?V_2\right).$$ (2) Show that $${\left(V_1?V_2\right)}^{?}?V_1^{?}?V_2^{?}$$. $$(3{)}^{?}$$ Show that $${\left(V_1?V_2\right)}^{?}?V_1^{?}?V_2^{?}$$. Recall that in the lecture we have show that the space of $$m\times n$$ matrices $${}_{{\text{Mat }}_{m\times n}}$$ can be identified with $${\mathbb{R}}^m?{\mathbb{R}}^n$$. The identification is given by $$E_{ij}?e_i?f_j$$ where $$E_{ij}$$ is the standard matrix basis, $$e_i$$ and $$f_j$$ are standard basis of $${\mathbb{R}}^m$$ and $${\mathbb{R}}^n$$. Recall a tensor in $$V_1?V_2$$ is called pure if it is equal to $$v_1?v_2$$ for some $$v_1?V_1$$ and $$v_2?V_2$$. (4) Using the above identification between $${\mathrm{Mat}}_{m\times n}$$ and $${\mathbb{R}}^m?{\mathbb{R}}^n$$ to show that a tensor in $${\mathbb{R}}^m?{\mathbb{R}}^n$$ is pure if and only if the corresponding matrix is of rank one. (5) Assume that $$m=n$$ in the pervious problem. Show that a 2-tensor in $${\mathbb{R}}^n?{\mathbb{R}}^n$$ is symmetric (resp. alternating) if and only if the corresponding matrix is symmetric (resp. skew symmetric).