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Thread: dimension of vector space

  1. #1
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    dimension of vector space

    Let $\displaystyle E,F,K$ be fields such that $\displaystyle F \subset E \subset K$. Prove that if $\displaystyle A=\{a_1,...,a_n\}$ is a basis for $\displaystyle K$ over $\displaystyle E$ and $\displaystyle B=\{b_1,...b_m\}$ is a basis for $\displaystyle E$ over $\displaystyle F$, then $\displaystyle C=\{a_ib_j, 1 \leq i \leq n, 1 \leq j \leq m\}$ is a basis for $\displaystyle K$ over $\displaystyle F$. Conclude that if $\displaystyle \dim_E(K)$ and $\displaystyle \dim_F(E)$ are finite, then $\displaystyle \dim_F(K)=\dim_E(K)\dim_F(E)$.

    I know how to do the proof, but I'm not sure how to conclude that $\displaystyle \dim_F(K)=\dim_E(K)\dim_F(E)$. Some help please.
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  2. #2
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    $\displaystyle \dim_E(K)=n,\dim_F(E)=m,\dim_F(K)=n*m,\longrightar row \dim_F(K)=\dim_E(K)\dim_F(E)\ ?
    $
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