Let V and W be vector spaces with subspaces V 1 andW 1, respectively. If T: V---
ID: 2940344 • Letter: L
Question
Let V and W be vector spaces with subspaces V1 andW1, respectively. If T: V--->W is linear,prove that T(V1) is a subspace of W and that {x V: T(x) W1} is a subspace of V.Anyone that can help me before the end of tonight, I will rateLifesaver!!!
Explanation / Answer
When you go about solving this equation. Always use the definitionsof what you are working for. Working with subspaces: When something is a subspace, they follow these properties: -It's non-empty (it's almost given every time) -A '0' element belongs in the subspace -The if 'a' and 'b' belong to S, then 'a+b' belongs to S -if 'a' belongs to S and 'c' is a scalar multiple, then 'c * a'belongs to S Lastly we need to look at the definition of LinearTransformation: Let T be a linear transformation from A to B, then T satisfiesthese properties: -T(a + b) = T(a) + T(b) for all elements a that belong to A and bthat belongs to B -T(ca) = c*T(a) for all elements a that belong to A and scalarmultiples c Finally we can move onto the actual problem itself. You are giventhat V and W are vector spaces following that A and B are subspacesof V and W respectively (keep in mind I am using notation a bitdifferently). Then they give that T is a linear transformation fromV ----> W. Want to show that T(A) is a subspace of W. Since A is a subspace, A is nonempty and contains the 0 zeroelement. From that, we can conclude there exists elements in A.T(A) is non empty. Keep in mind that T(A) is an element in W sinceT: V -----> W Second, we can take arbitrary elements in A: namely v_1 andv_2. T(v_1 + v_2) = T(v_1) + T(v_2) (given that T is a lineartransformation) T(v_1) and T(v_2) are both elements in W (since T maps the elementsinto W) T(v_1) = w_1 and T(v_2) = w_2 for some elements in V and W T(v_1) + T(v_2) = w_1 + w_2 Since W is a vector space, w_1 + w_2 also belongs to W. Lastly, we need to show scalar multiples. We know that since A is a subspace and v_1 belongs to A. c*v_1 alsobelongs to A for all scalars c. We want to look at T(c*v_1) = c*T(v_1) since T is a lineartransformation c*T(v_1) = c* w_1 for some arbitrary vectors in V and W. Since W is a vector space, c*w_1 belongs to W. You can use the addition of two arbitrary vectors to show that thezero element belongs into W too. Thus T(A) is a subspace of W. Unfortunately I have to go, but for the second part it gives youthat T(x) are elements of B. Using similar configuration, you wantto show that W_1 is a subset of V. Once again using the definitionsof subspaces/vectorspaces and linear transformation you canconclude that the properties carry over and not only is W_1 asubset of V, but also a subspace.
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