Question 2 Suppose that T :V +V is a linear transformation. Further suppose {01, v2} forms...
Let T: V + W be a linear transformation. Assume that T is one-to-one. Prove that if {V1, V2, V3} C V is a linearly independent subset of V, then {T(01), T(v2), T(13)} C W is a linearly independent subset of W.
7. Let T : V → W be a linear transformation, and let v1,v2,...,vn be vectors in V. Suppose that T (v1), T (v2), . . . , T (vn) are linearly independent. Show that v1, v2, . . . , vn are linearly independent.
2. Suppose that V is an inner product space. (i) Prove that, for any vectors 01, 02 € V, || 0111? + || 0,2||2 = || v1 + v2||2 + || 01 – v2||2 2 (ii) Prove that, for any vectors V1, V2 € V, if v, and v, are orthogonal then || 01 || + || 112 || 2 = || 01 + 02||2.
3. Let V be a finite dimensional inner product space, and suppose that T is a linear operator on this space. (i) Let B be an ordered orthonormal basis for V and let U be the linear operator on V determined by [U19 = (T);. Then, for all 01,09 € V, (01, T(02)) = (U(V1), v2) (ii) Prove that the conclusion of the previous part does not hold, in general, if the basis 8 is not orthonormal.
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...
Problem 6: Let B = {V1, V2, ..., Un} be a set of vectors in R", and let T:R" → R" be a linear transformation such that the set {T(01), T(V2), ...,T(Un) } is basis for R". Show that B = {01, V2, ..., Un } is also a basis for R". Problem 7: Decide whether the following statement is true or false. If it is true, prove it. If it is false, give an example to show that it...
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,...
Question 8 [10 points] Suppose T: RM22 is a linear transformation whose action on a basis for R4 is as follows -1 1 -11 4 4 0 1 1 0 1 -1 1 -45 1 2-2 1 -1 7 0 Determine whether T is one-to-one andlor onto. If it is not one-to-one, show this by providing two vectors that have the same image under T. If T is yt onto, show this by providing a matrix in M22 that is...
Question 2. a) The zero transformation. We define the zero transformation, To: FN → Fm by To(x) = 0 VxEFN. (i) What is R(To)? (ii) Is To onto? (iii) What is N(To)? (iv) Is To one-to-one? (v) What is (To]s? b) The identity transformation. We define the identity transformation, Tj: Fn + En by Ty(x) = x V xEFN. (i) What is R(Ti)? (ii) Is T, onto? (iii) What is N(T)? (iv) Is T one-to-one? (v) What is Ti]s? Question...
7.) 10points Let V be the space of 2 x 2 matrices. Let T: V-V be given by T(A) = A a.) Prove that T a linear transformation b.) Find a basis for the nullspace (Kernel) of T. c) Find a basis for the range of T. 7.) 10points Let V be the space of 2 x 2 matrices. Let T: V-V be given by T(A) = A a.) Prove that T a linear transformation b.) Find a basis for...