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Thread: Prove that the inverse image of V is a subspace in X.

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    Prove that the inverse image of V is a subspace in X.

    Prove that if $\displaystyle f$ is a linear transformation from a vector space $\displaystyle X$ to a vector space $\displaystyle Y$, then for any subspace $\displaystyle V$ in $\displaystyle Y$, $\displaystyle f^{-1}[V]$ is a subspace in $\displaystyle X$.
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    Quote Originally Posted by mathwizard View Post
    Prove that if $\displaystyle f$ is a linear transformation from a vector space $\displaystyle X$ to a vector space $\displaystyle Y$, then for any subspace $\displaystyle V$ in $\displaystyle Y$, $\displaystyle f^{-1}[V]$ is a subspace in $\displaystyle X$.
    Let $\displaystyle \bold{x},\bold{y}\in f^{-1}[V]$ and $\displaystyle a\in F$, the base field for the vector spaces. It means by definition that $\displaystyle f(\bold{x}),f(\bold{y}) \in V$. Since $\displaystyle V$ is a subspace it means $\displaystyle f(\bold{x})+f(\bold{y}) \in V$. Therefore, since $\displaystyle f$ is a linear transformation it means $\displaystyle f(\bold{x}+\bold{y}) \in V$ which means $\displaystyle \bold{x}+\bold{y} \in f^{-1}[V]$. Thus, $\displaystyle f^{-1}[V]$ is closed under vector addition. Likewise, $\displaystyle kf(\bold{x}) \in V $ which means $\displaystyle k\bold{x}\in f^{-1}[V]$. Thus, $\displaystyle f^{-1}[V]$ is closed under scalar multiplications. All the other properties for being a vector space are satisfied because $\displaystyle f^{-1}[V]\subseteq X$. Thus, $\displaystyle f^{-1}[V]$ is a vector space over $\displaystyle F$.
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    Quote Originally Posted by ThePerfectHacker View Post
    Let $\displaystyle \bold{x},\bold{y}\in f^{-1}[V]$ and $\displaystyle a\in F$, the base field for the vector spaces. It means by definition that $\displaystyle f(\bold{x}),f(\bold{y}) \in V$. Since $\displaystyle V$ is a subspace it means $\displaystyle f(\bold{x})+f(\bold{y}) \in V$. Therefore, since $\displaystyle f$ is a linear transformation it means $\displaystyle f(\bold{x}+\bold{y}) \in V$ which means $\displaystyle \bold{x}+\bold{y} \in f^{-1}[V]$. Thus, $\displaystyle f^{-1}[V]$ is closed under vector addition. Likewise, $\displaystyle kf(\bold{x}) \in V $ which means $\displaystyle k\bold{x}\in f^{-1}[V]$. Thus, $\displaystyle f^{-1}[V]$ is closed under scalar multiplications. All the other properties for being a vector space are satisfied because $\displaystyle f^{-1}[V]\subseteq X$. Thus, $\displaystyle f^{-1}[V]$ is a vector space over $\displaystyle F$.
    Thanks for your input; your proof confirms my belief that the same proof in my linear algebra textbook is incorrect; it messed up on x & y and f(x) & f(y) (and it wasted me a long time to try to make sense out of that flawed proof ).

    PS: I think there's one part that your proof is incomplete; it hasn't showed that the zero vector is in $\displaystyle f^{-1}[V].$
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    Quote Originally Posted by mathwizard View Post
    PS: I think there's one part that your proof is incomplete; it hasn't showed that the zero vector is in $\displaystyle f^{-1}[V].$
    Yes it has. If $\displaystyle \bold{x}\in f^{-1}[V]$ we proved $\displaystyle k\bold{x} \in f^{-1}[V]$. Now let $\displaystyle k=0$.
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    Quote Originally Posted by ThePerfectHacker View Post
    Yes it has. If $\displaystyle \bold{x}\in f^{-1}[V]$ we proved $\displaystyle k\bold{x} \in f^{-1}[V]$. Now let $\displaystyle k=0$.
    Using your logic, why must there be a condition that a subspace must contain a zero vector, since one of the other two conditions (i.e., closed under scalar multiplication) has already taken care of the first condition (by letting k=0 as you said)?
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    Quote Originally Posted by mathwizard View Post
    why must there be a condition that a subspace must contain a zero vector, since one of the other two conditions (i.e., closed under scalar multiplication) has already taken care of the first condition (by letting k=0 as you said)?
    The condition is necessary, in order to exclude the possibility of the set being empty (the empty set is by convention not considered to be a subspace). If you're trying to show that X is a subspace then you can only make use of the implication $\displaystyle \bold{x}\in X\Rightarrow 0\bold{x}\in X\Rightarrow \bold{0}\in X$ if you know that there exists a vector x in X.
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