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Question# 5-8

Question# 5-8

Advanced Engineering Mathematics
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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Question# 5-8

Th 3.3 i): Let T V W be linear with r = rank (T). Let {w₁,..., wr} be a basis for Rg(T). Let U for i = 1,..., r be any choice of elements of V satis- fying T(v;) =w₁, Vi = 1,..., r. Let v = nullity (T) and let {v₁,..., v₂} be a basis for Ker(T). Then {₁,..., Ur, U₁,...,U} is a basis for V. 5. Apply Theorem 3.3 i) to V = R, W = Rm and T(x) = Ax, where Amxn to find a basis for R" pro- duced by applying Gaussian elimination to A (Hint: This is in the notes). Explain your answer. 6. Apply Theorem 3.3 i) to V = M(n, R), W = R and T(A) = Tr(A), to show that a basis for sl(n, R) augmented by In is a basis for M(n, R). (Hint: This is in the notes). Explain your answer. 7. What basis for V = M(n, R) does Theorem 3.3 i) yield when applied to T(A) = A + AT, with W M(n, R), when using the standard basis = {eje + eje}<i<j<n U {e;e?}=1...n for symmetric matrices (Hint: This too is in the notes). Explain your answer. 8. Apply Theorem 3.3 i) to find a basis for V = P₁ with when using the usual basis, W = P₁-1 {1, x,x²,...,x-¹}, for Pn-1. Explain your answer. and T (p) = = 7
Transcribed Image Text:Th 3.3 i): Let T V W be linear with r = rank (T). Let {w₁,..., wr} be a basis for Rg(T). Let U for i = 1,..., r be any choice of elements of V satis- fying T(v;) =w₁, Vi = 1,..., r. Let v = nullity (T) and let {v₁,..., v₂} be a basis for Ker(T). Then {₁,..., Ur, U₁,...,U} is a basis for V. 5. Apply Theorem 3.3 i) to V = R, W = Rm and T(x) = Ax, where Amxn to find a basis for R" pro- duced by applying Gaussian elimination to A (Hint: This is in the notes). Explain your answer. 6. Apply Theorem 3.3 i) to V = M(n, R), W = R and T(A) = Tr(A), to show that a basis for sl(n, R) augmented by In is a basis for M(n, R). (Hint: This is in the notes). Explain your answer. 7. What basis for V = M(n, R) does Theorem 3.3 i) yield when applied to T(A) = A + AT, with W M(n, R), when using the standard basis = {eje + eje}<i<j<n U {e;e?}=1...n for symmetric matrices (Hint: This too is in the notes). Explain your answer. 8. Apply Theorem 3.3 i) to find a basis for V = P₁ with when using the usual basis, W = P₁-1 {1, x,x²,...,x-¹}, for Pn-1. Explain your answer. and T (p) = = 7
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