Homework Answers
Please give answer with the details. Thanks a lot! Let T: V-W be a linear transformation between vector spaces V and W...
Please give answer with the details. Thanks a lot!
Let T: V-W be a linear transformation between vector spaces V and W (1) Prove that if T is injective (one-to-one) and {vi,.. ., vm) is a linearly independent subset of V the n {T(6),…,T(ền)} is a linearly independent subset of W (2) Prove that if the image of any linearly independent subset of V is linearly independent then Tis injective. (3) Suppose that {b1,... bkbk+1,. . . ,b,) is a...
(4) Let V and W be vector spaces over R: consider the free vector space F(V × W) on the Cartesian...
(4) Let V and W be vector spaces over R: consider the free vector space F(V × W) on the Cartesian product V x W of V and W. Given an element (v, w) of V x W, we view (v, w) as an element of F(V x W) via the inclusion map i : V x W F(V x W) Any element of F(V x W) is a finite linear combination of such elements (v, w) Warning. F(V ×...
Problem 5. Let V and W be vector spaces, and suppose that B (vi, ..., Vn) is a basis of V a) Prove that for every funct...
Problem 5. Let V and W be vector spaces, and suppose that B (vi, ..., Vn) is a basis of V a) Prove that for every function f : B → W, there exists a linear transformation T: V → W such that T(v;)-f(7) for all vEB (b) Prove that for any two linear transformations S : V → W and T : V → W, if S(6) = T(6) for all ï, B, then S = T (c) Prove...
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0},...
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0}, where w.v is the dot product. 2 (a) Prove that U is a subspace of R3. (b) Find a basis for U and compute its dimension. 4. Decide whether or not the following subsets of vector spaces are linearly independent. If they are, prove it. If they aren't, write one as a linear combination of the others. (a) The subset {0 0 0 of...
Let V and W be finite dimensional vector spaces and let T:V → W be a...
Let V and W be finite dimensional vector spaces and let T:V → W be a linear transformation. We say a linear transformation S :W → V is a left inverse of T if ST = Iy, where Iy denotes the identity transformation on V. We say a linear transformation S:W → V is a right inverse of T if TS = Iw, where Iw denotes the identity transformation on W. Finally, we say a linear transformation S:W → V...
Let V and W be two vector spaces over R and T:V + W be a...
Let V and W be two vector spaces over R and T:V + W be a linear transformation. We call a linear map S: W → V a generalized inverse of T if To SOT = T and SoTo S = S. If V and W are finite dimensional, show that there exists a generalized inverse of T.
Problem 5: Let V and W be vector spaces and let B = {V1, V2, ...,...
Problem 5: Let V and W be vector spaces and let B = {V1, V2, ..., Un} CV be a basis for V. Let L :V + W be a linear transformation, and let Ker L = {2 € V: L(x)=0}. (a) If Ker L = {0}, show that C = {L(v1), L(02), ..., L(vn) } CW is a linearly independent set in W. (b) If C = {L(01), L(V2),..., L(Un)} C W is a linearly independent set in W,...
Let V and W be two vector spaces over R and T:V + W be a...
Let V and W be two vector spaces over R and T:V + W be a linear transformation. We call a linear map S: W → V a generalized inverse of T if To SOT = T and SoTo S = S. If T is an isomorphism, show that T-1 is the unique generalized inverse of T.
Let V and W be two vector spaces over R and T:V + W be a...
Let V and W be two vector spaces over R and T:V + W be a linear transformation. We call a linear map S:W → V a generalized inverse of T if To SoT=T and SoToS = S. If V and W are finite dimensional, show that there exists a generalized inverse of T.
Please give answer with the details. Thanks a lot!
Let T: V-W be a linear transformation between vector spaces V and W (1) Prove that if T is injective (one-to-one) and {vi,.. ., vm) is a linearly independent subset of V the n {T(6),…,T(ền)} is a linearly independent subset of W (2) Prove that if the image of any linearly independent subset of V is linearly independent then Tis injective. (3) Suppose that {b1,... bkbk+1,. . . ,b,) is a...
(4) Let V and W be vector spaces over R: consider the free vector space F(V × W) on the Cartesian product V x W of V and W. Given an element (v, w) of V x W, we view (v, w) as an element of F(V x W) via the inclusion map i : V x W F(V x W) Any element of F(V x W) is a finite linear combination of such elements (v, w) Warning. F(V ×...
Problem 5. Let V and W be vector spaces, and suppose that B (vi, ..., Vn) is a basis of V a) Prove that for every function f : B → W, there exists a linear transformation T: V → W such that T(v;)-f(7) for all vEB (b) Prove that for any two linear transformations S : V → W and T : V → W, if S(6) = T(6) for all ï, B, then S = T (c) Prove...
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0}, where w.v is the dot product. 2 (a) Prove that U is a subspace of R3. (b) Find a basis for U and compute its dimension. 4. Decide whether or not the following subsets of vector spaces are linearly independent. If they are, prove it. If they aren't, write one as a linear combination of the others. (a) The subset {0 0 0 of...
Let V and W be finite dimensional vector spaces and let T:V → W be a linear transformation. We say a linear transformation S :W → V is a left inverse of T if ST = Iy, where Iy denotes the identity transformation on V. We say a linear transformation S:W → V is a right inverse of T if TS = Iw, where Iw denotes the identity transformation on W. Finally, we say a linear transformation S:W → V...
Let V and W be two vector spaces over R and T:V + W be a linear transformation. We call a linear map S: W → V a generalized inverse of T if To SOT = T and SoTo S = S. If V and W are finite dimensional, show that there exists a generalized inverse of T.
Problem 5: Let V and W be vector spaces and let B = {V1, V2, ..., Un} CV be a basis for V. Let L :V + W be a linear transformation, and let Ker L = {2 € V: L(x)=0}. (a) If Ker L = {0}, show that C = {L(v1), L(02), ..., L(vn) } CW is a linearly independent set in W. (b) If C = {L(01), L(V2),..., L(Un)} C W is a linearly independent set in W,...
Let V and W be two vector spaces over R and T:V + W be a linear transformation. We call a linear map S: W → V a generalized inverse of T if To SOT = T and SoTo S = S. If T is an isomorphism, show that T-1 is the unique generalized inverse of T.
Let V and W be two vector spaces over R and T:V + W be a linear transformation. We call a linear map S:W → V a generalized inverse of T if To SoT=T and SoToS = S. If V and W are finite dimensional, show that there exists a generalized inverse of T.
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