How do I prove that a spanning set when transformed spans the range of a linear map T?
up vote
0
down vote
favorite
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
add a comment |
up vote
0
down vote
favorite
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
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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
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 at 5:15
Jaigus
2308
2308
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
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.
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
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.
add a comment |
up vote
0
down vote
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.
add a comment |
up vote
0
down vote
up vote
0
down vote
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.
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 at 5:25
DeepSea
70.6k54487
70.6k54487
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3004551%2fhow-do-i-prove-that-a-spanning-set-when-transformed-spans-the-range-of-a-linear%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown