How do I prove that a spanning set when transformed spans the range of a linear map T?











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I started out with the following reasoning: Consider a spanning set $v in V: v_1, ..., v_n$. By definition every element in $V$ is some linear combination of this list. By linearity, $T(alpha v_1 + ... + alpha v_n)$ = $alpha T(v_1) + ... +alpha T(v_n)$, which belongs to the Range of T.



I'm not sure how to finish this and put everything together.






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    I started out with the following reasoning: Consider a spanning set $v in V: v_1, ..., v_n$. By definition every element in $V$ is some linear combination of this list. By linearity, $T(alpha v_1 + ... + alpha v_n)$ = $alpha T(v_1) + ... +alpha T(v_n)$, which belongs to the Range of T.



    I'm not sure how to finish this and put everything together.






    linear-algebra linear-transformations







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      I started out with the following reasoning: Consider a spanning set $v in V: v_1, ..., v_n$. By definition every element in $V$ is some linear combination of this list. By linearity, $T(alpha v_1 + ... + alpha v_n)$ = $alpha T(v_1) + ... +alpha T(v_n)$, which belongs to the Range of T.



      I'm not sure how to finish this and put everything together.






      linear-algebra linear-transformations







      share|cite|improve this question













      I started out with the following reasoning: Consider a spanning set $v in V: v_1, ..., v_n$. By definition every element in $V$ is some linear combination of this list. By linearity, $T(alpha v_1 + ... + alpha v_n)$ = $alpha T(v_1) + ... +alpha T(v_n)$, which belongs to the Range of T.



      I'm not sure how to finish this and put everything together.






      linear-algebra linear-transformations




      linear-algebra linear-transformations






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      Jaigus

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          Let $w in text{Ran}{(T)}implies w = T(u), u in Vimplies u = a_1v_1+a_2v_2+cdots+a_nv_nimplies w = T(u) = a_1T(v_1)+a_2T(v_2)+cdots +a_nT(v_n)$. This implies the claim.






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            Let $w in text{Ran}{(T)}implies w = T(u), u in Vimplies u = a_1v_1+a_2v_2+cdots+a_nv_nimplies w = T(u) = a_1T(v_1)+a_2T(v_2)+cdots +a_nT(v_n)$. This implies the claim.






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              Let $w in text{Ran}{(T)}implies w = T(u), u in Vimplies u = a_1v_1+a_2v_2+cdots+a_nv_nimplies w = T(u) = a_1T(v_1)+a_2T(v_2)+cdots +a_nT(v_n)$. This implies the claim.






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                Let $w in text{Ran}{(T)}implies w = T(u), u in Vimplies u = a_1v_1+a_2v_2+cdots+a_nv_nimplies w = T(u) = a_1T(v_1)+a_2T(v_2)+cdots +a_nT(v_n)$. This implies the claim.






                share|cite|improve this answer












                Let $w in text{Ran}{(T)}implies w = T(u), u in Vimplies u = a_1v_1+a_2v_2+cdots+a_nv_nimplies w = T(u) = a_1T(v_1)+a_2T(v_2)+cdots +a_nT(v_n)$. This implies the claim.







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                answered 10 hours ago









                DeepSea

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