How do I prove that a spanning set when transformed spans the range of a linear map T?
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
asked Nov 19 '18 at 5:15
Jaigus
2218
add a comment |
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
asked Nov 19 '18 at 5:15
Jaigus
2218
add a comment |
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
asked Nov 19 '18 at 5:15
Jaigus
2218
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
asked Nov 19 '18 at 5:15
Jaigus
2218
asked Nov 19 '18 at 5:15
Jaigus
2218
asked Nov 19 '18 at 5:15
Jaigus
2218
asked Nov 19 '18 at 5:15
Jaigus
2218
asked Nov 19 '18 at 5:15
Jaigus
2218
2218
add a comment |
add a comment |
1 Answer
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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.
answered Nov 19 '18 at 5:25
DeepSea
70.9k54487
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1 Answer
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1 Answer
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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.
answered Nov 19 '18 at 5:25
DeepSea
70.9k54487
add a comment |
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.
answered Nov 19 '18 at 5:25
DeepSea
70.9k54487
add a comment |
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.
answered Nov 19 '18 at 5:25
DeepSea
70.9k54487
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.
answered Nov 19 '18 at 5:25
DeepSea
70.9k54487
answered Nov 19 '18 at 5:25
DeepSea
70.9k54487
answered Nov 19 '18 at 5:25
DeepSea
70.9k54487
answered Nov 19 '18 at 5:25
DeepSea
70.9k54487
70.9k54487
add a comment |
add a comment |
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