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Math Help - isomorphism

  1. #1
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    isomorphism

    if t is an isomorphism of V onto W,prove that t maps a basis of V onto a basis of W
    thanks
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by prashantgolu View Post
    if t is an isomorphism of V onto W,prove that t maps a basis of V onto a basis of W
    thanks
    I'll let T:V\to W is the isomorphism, and \{v_1,\cdots,v_n\} a basis for V. Note that since T is injective that \ker T=\{\bold{0}\} (why?) and so \displaystyle \sum_{j=1}^{n}\alpha_j T\left(v_j\right)=\bold{0}\implies T\left(\sum_{j=0}^{n}\alpha_j v_j\right)=\bold{0}\implies \sum_{j=1}^{n}\alpha_j v_j=\bold{0}\implies \alpha_1=\cdots=\alpha_n and so \left\{T(v_1),\cdots,T(v_n)\right\} is l.i. Now, since T is surjective we have that for any w\in W there exists some v\in V such that T(v)=w. Note though that since \{v_1,\cdots,v_n\} is a basis for there exists \alpha_1,\cdots,\alpha_n\in F such that \displaystyle \sum_{j=1}^{n}\alpha_j v_j=v and so \displaystyle w=T(v)=T\left(\sum_{j=1}^{n}\alpha_j v_j\right)=\sum_{j=1}^{n}\alpha_j T(v_j) and thus \left\lange\left\{T(v_1),\cdots,T(v_n)\right\}\rig  ht\rangle=W. Thus, \left\{T(v_1),\cdots,T(v_n)\right\} is a basis for W
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