math121b:final-sol
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math121b:final-sol [2020/05/18 00:27] pzhou |
math121b:final-sol [2026/02/21 14:41] (current) |
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| - (T) Given a basis $e_1, \cdots, e_n$ of $V$, there exists a basis $E^1, \cdots, E^n$ on $V^*$, such that $E^i (e_j) = \delta_{ij}$. | - (T) Given a basis $e_1, \cdots, e_n$ of $V$, there exists a basis $E^1, \cdots, E^n$ on $V^*$, such that $E^i (e_j) = \delta_{ij}$. | ||
| - (F) If we change $e_1$ in the basis $e_1, \cdots, e_n$, in the dual basis< | - (F) If we change $e_1$ in the basis $e_1, \cdots, e_n$, in the dual basis< | ||
| + | * The correct statement would be, "If we change $e_1$ in the basis $e_1, \cdots, e_n$, in the dual basis, not only $E_1$ will change, all other $E_i$ might also change." | ||
| - (F) Given a vector space with inner product, there exists a < | - (F) Given a vector space with inner product, there exists a < | ||
| - (T) Let $V$ and $W$ be two vector spaces with inner products and of the same dimension.Then there exists a linear map $f: V \to W$, such that for any $v_1, v_2 \in V$, $$\la v_1, v_2 \ra = \la f(v_1), f(v_2) \ra.$$ | - (T) Let $V$ and $W$ be two vector spaces with inner products and of the same dimension.Then there exists a linear map $f: V \to W$, such that for any $v_1, v_2 \in V$, $$\la v_1, v_2 \ra = \la f(v_1), f(v_2) \ra.$$ | ||
math121b/final-sol.1589761626.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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