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Math Help - prove T(V) is a subspace

  1. #1
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    prove T(V) is a subspace

    Let T : R^n --> R^mbe a linear transformation, if V is a subspace of R^n and
    T(V) = {T(v): v \in R^n},
    prove that T(V) is a subspace of R^m.
    Sry, I know this is an easy question, but i am kindof confused with the definition of subspace
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  2. #2
    Member Black's Avatar
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    Refer to this thread.
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  3. #3
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    so is this right

    If v, w \in V, then T(v) = v and T(w) = w so we get
    T(v + w) = T(v) + T(w) = v + w
    which means that v+w \in V and condition (1) holds.
    For any scalar c,
    T(cv) = cT(v) = cv
    so that cv \in Vand condition (2) holds so V is a subspace

    is R^msame as R^n?
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  4. #4
    Member Black's Avatar
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    Quote Originally Posted by wopashui View Post
    so is this right

    If v, w \in V, then T(v) = v and T(w) = w so we get
    T(v + w) = T(v) + T(w) = v + w
    which means that v+w \in V and condition (1) holds.
    For any scalar c,
    T(cv) = cT(v) = cv
    so that cv \in Vand condition (2) holds so V is a subspace

    is R^msame as R^n?
    Not in general. If n=m, then \mathbb{R}^n=\mathbb{R}^m ; they are not the same otherwise.

    To show that T(V) is a subspace of \mathbb{R}^m, you have to prove three conditions:

    (1) 0_{\mathbb{R}^m} \in T(V),
    (2) If u,v \in T(V), then u+v \in T(V),
    (3) If \lambda \in \mathbb{R} and u \in T(V), then \lambda u \in T(V).

    For (1), 0_{\mathbb{R}^m}=T\left(0_{\mathbb{R}^n}\right) \in T(V).

    For (2), if u,v \in T(V), then pick u_0,v_0 \in \mathbb{R}^n such that T(u_0)=u and T(v_0)=v. Then

    u+v=T(u_0)+T(v_0)=T(u_0+v_0) \in T(V).

    And for (3), you just use preservation of scalar multiplication.
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  5. #5
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    Quote Originally Posted by Black View Post
    Not in general. If n=m, then \mathbb{R}^n=\mathbb{R}^m ; they are not the same otherwise.

    To show that T(V) is a subspace of \mathbb{R}^m, you have to prove three conditions:

    (1) 0_{\mathbb{R}^m} \in T(V),
    (2) If u,v \in T(V), then u+v \in T(V),
    (3) If \lambda \in \mathbb{R} and u \in T(V), then \lambda u \in T(V).

    For (1), 0_{\mathbb{R}^m}=T\left(0_{\mathbb{R}^n}\right) \in T(V).

    For (2), if u,v \in T(V), then pick u_0,v_0 \in \mathbb{R}^n such that T(u_0)=u and T(v_0)=v. Then

    u+v=T(u_0)+T(v_0)=T(u_0+v_0) \in T(V).

    And for (3), you just use preservation of scalar multiplication.
    so for (3) if u\in T(v), c \in R, then we have
    cu = cT(u') \in T(v)

    right?
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  6. #6
    Member Black's Avatar
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    Quote Originally Posted by wopashui View Post
    so for (3) if u\in T(v), c \in R, then we have
    cu = cT(u') = T(cu') \in T(v)

    right?
    Need to add that part.
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  7. #7
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    Quote Originally Posted by Black View Post
    Need to add that part.
    You certainly provided a very good description. Thanks a lot.
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