(7) Let ? and ? be vector spaces and ? a subspace of ? . Suppose ? : ? → ? is a linear ̃︀ ̃︀ transformation with ???(?) = ? and deﬁne ? : ? ∖ ? → ? by ?(? + ?) := ?(?). ̃︀ Prove ? is an isomorphism from ? ∖ ? to ??(?), that is ? ∖ ???(?) ≃ ??(?). (In abstract algebra, this is known as the ﬁrst isomorphism theorem!) (7) Let \( V \) and \( U \) be vector spaces and \( W \) a subspace of \( V \). Suppose \( T: V \to U \) is a linear transformation with \( \text{ker}(T) = W \) and define \( \widetilde{T}: V / W \to U \) by \( \widetilde{T}(v + W) := T(v) \). Prove \( \widetilde{T} \) is an isomorphism from \( V / W \) to \( \text{Im}(T) \), that is \( V / \text{ker}(T) \cong \text{Im}(T) \). (In abstract algebra, this is known as the first isomorphism theorem!)

Answered: Image /qna-images/answer/28920ac0-384b-407f-a19b-13f8bb74684c.jpg

(7) Let V and U be vector spaces and W a subspace of V. Suppose T : V → U is a linear transformation with ker(T) = W and define T : V \W →U by T(v + W) := T(v). Prove T is an isomorphism from V \ W to Im(T), that is V \ ker(T) ~ Im(T). (In abstract algebra, this is known as the first isomorphism theorem!)

(7) Let V and U be vector spaces and W a subspace of V. Suppose T : V → U is a linear transformation with ker(T) = W and define T : V \W →U by T(v + W) := T(v). Prove T is an isomorphism from V \ W to Im(T), that is V \ ker(T) ~ Im(T). (In abstract algebra, this is known as the first isomorphism theorem!)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Consider the following three transformations from R³ to R³: x1 T x2 x1 - 2x3 2x1 - 2x2 - 5x3 x1 x1+5 x1 S x2 x2 R x2 = 0 X3 -x1 + 2x2+8x3) x3 x1x3 X3 Which of these transformations is/are linear? Select all correct choices. T S R None of these
+b] Let T : R? → R be a linear transformation. Which of the following is equal to T a +b O A. T (B) +" (!) В. T +T +T +T Oc. T 피 (B) OD. | 2T
2. trace. Let n = 4 (we choose 4 for calculation simplicity, but the following are true for arbitrary n). Consider the following linear transformation: tr : R" +R a11 a12 a13 ain ... a22 a23 a2n ... a21 a32 a33 Ha11 + a22 + a33 + ...+ ann = ... a31 i=1 An3 ann ... anl an2 Let In be the n x n identity matrix (or equivalently, a vector in R""). Calculate (A) that tr(In) = n. Let A, B be two n x n matrices (or equivalently, a vector in Rn*). Calculate that (B) tr(AB) = tr(BA). Hint. Write A = [aj] and B = [br1]. Write down the diagonal of AB and BA in terms of aij and bri. Then compute tr(AB) and tr(BA).
38. a. Consider a linear transformation T from R³ to R. What are the possible values of dim(ker T)? Explain. b. Consider a linear transformation T from R4 to R'. What are the possible values of dim(imT)? Explain.
. Tell whether the following sets are open: (Justify your answer.) (a) (3,5) U {6} (b) (-∞, 0) U (0, 1) (c) {1, 2, 3, 4, 5, 6, 7, 8, 9) (d) (-∞, 0) U[0, 1) (e) Z (f) (-∞,0) U (0, 1] (h) R - {1, 2, 3} (j) { //: neN} U {0} (1) Qn (0,1) (g) (-∞,0) U [0, 1] (i) { //:ne N} (k) Q
GIVEN: P₂ = { a₁ + a₁x + a₂x² ao, a₁, a₂ ≤ R} T: P₂ R, T(a₁ + a₁x + a₂x²) = a₂ For example: 7(1- x + 3x²) = 3 PROVE: T is a linear transformation. -
Is T linear ? Is T one-to-one? Is T onto?
Suppose that T: R³ R³ is a linear transformation satisfying T xpression below (DO NOT ROUND) 0-0-0-0-0-0 ---A Then T 1 and T