Linear Algebra Exercises

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Problem 1: (A) Showπ = {[ π π ]βΆπ+π+π=0, π,π,π,πβπΉ}isasubspaceofπ22(πΉ) π+π πβπ (B) i) Findthebasisanddimensionforπ={[ π π ]βΆπ+π+π=0, π,π,π,πβπΉ} π+π πβπ ii) Explain why ππππ{π} β  π22(πΉ) 2 Problem 2: 3 (A) (B) (C) i) ii) Let π΅ = {1 + π₯ + π₯2, 1 β π₯ β π₯2, 1 β π₯ + π₯2} be a basis for π (πΉ). 2 1 Determine π€ when [π€]π΅ = [ 0 ] β2 Determine [π¦]π΅ when π¦ = 4 + 2π₯. Let π΅ = {1 + π₯ + π₯2, 1 β π₯ β π₯2, 1 β π₯ + π₯2} and πΆ = {1 + π₯, 2 β π₯2, π₯2}. Find State the expression that relates [π£]π΅ with [π£]πΆ and π . πΆβπ΅ π . πΆβπ΅ Problem 4: 5 (A) i) ii) iii) iv) Let π΅ = {[1 0],[0 1],[0 0],[0 0]} be a basis for π22(πΉ) and πΆ = {1,π₯,π₯2,π₯3} 00001001 be a basis for π (πΉ). Let T be a linear transformation given by: 3 π(π π)=ππ₯3 +(πβπ+π)π₯2 +(π+2π)π₯+(πβπ) ππ Find the matrix representation of transformation π with respect to basis π΅ β πΆ. Use part i) to find the matrix representation of transformation πβ1 from πΆ β π΅. Use part ii) to find the inverse transformation πβ1(π + ππ₯ + ππ₯2 + ππ₯3) Is π injective, surjective, both, or neither. How do you know? Problem 6: (A) i) ii) 011 Consider a basis π΅ = {[ 0 ],[ 0 ],[β1]} for the space ππππ{π΅}. β1 β1 3 0 1 β1 Use the Gram Schmidt Process on π΅ to find an orthonormal basis for ππππ{π΅}. Let matrix π΄ be created using the vectors in π΅ as the columns of π΄. Find the ππ factorization for matrix π΄. 7

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