2. (a) Let V be an n-dimensional vector space, and let {v₁, ..., Un} be a basis for V.Show that if T: V → V is a linear transformation such that T(v₁) = 0 andT(v₁) € span{₁,..., Vi-1} for all i = {2,3,...,n}, then 7" = 0.(b) Use (a) to show that if A € Mnxn(F) is a strictly upper triangular matrix(meaning that all entries on or below the main diagonal are 0), then A = 0.

Given V is n- dimensional vector space and v1,…,vn is a basis for V.

2. (a) Let V be an n-dimensional vector space, and let {₁,...,Vn} be a basis for V. Show that if T: V → V is a linear transformation such that T(v₁) 0 and T(vi) Espan{v₁,..., Vi-1} for all i = {2, 3,...,n}, then T = 0. = (b) Use (a) to show that if A € Mnxn (F) is a strictly upper triangular matrix (meaning that all entries on or below the main diagonal are 0), then A" = 0.

2. (a) Let V be an n-dimensional vector space, and let {₁,...,Vn} be a basis for V. Show that if T: V → V is a linear transformation such that T(v₁) 0 and T(vi) Espan{v₁,..., Vi-1} for all i = {2, 3,...,n}, then T = 0. = (b) Use (a) to show that if A € Mnxn (F) is a strictly upper triangular matrix (meaning that all entries on or below the main diagonal are 0), then A" = 0.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2. (a) Let V be an n-dimensional vector space, and let {v₁, ..., Un} be a basis for V.
Show that if T: V → V is a linear transformation such that T(v₁) = 0 and
T(v₁) € span{₁,..., Vi-1} for all i = {2,3,...,n}, then 7" = 0.
(b) Use (a) to show that if A € Mnxn(F) is a strictly upper triangular matrix
(meaning that all entries on or below the main diagonal are 0), then A = 0.

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