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Problem 1: (A) Show𝑆 = {[ π‘Ž π‘Ž ]βˆΆπ‘Ž+𝑏+𝑐=0, π‘Ž,𝑏,𝑐,π‘‘βˆˆπ‘Ή}isasubspaceof𝑀22(𝑹) 𝑏+𝑑 π‘βˆ’π‘‘ (B) i) Findthebasisanddimensionfor𝑆={[ π‘Ž π‘Ž ]βˆΆπ‘Ž+𝑏+𝑐=0, π‘Ž,𝑏,𝑐,π‘‘βˆˆπ‘Ή} 𝑏+𝑑 π‘βˆ’π‘‘ ii) Explain why π‘†π‘π‘Žπ‘›{𝑆} β‰  𝑀22(𝑹) 2 Problem 2: 3 (A) (B) (C) i) ii) Let 𝐡 = {1 + π‘₯ + π‘₯2, 1 βˆ’ π‘₯ βˆ’ π‘₯2, 1 βˆ’ π‘₯ + π‘₯2} be a basis for 𝑃 (𝑹). 2 1 Determine 𝑀 when [𝑀]𝐡 = [ 0 ] βˆ’2 Determine [𝑦]𝐡 when 𝑦 = 4 + 2π‘₯. Let 𝐡 = {1 + π‘₯ + π‘₯2, 1 βˆ’ π‘₯ βˆ’ π‘₯2, 1 βˆ’ π‘₯ + π‘₯2} and 𝐢 = {1 + π‘₯, 2 βˆ’ π‘₯2, π‘₯2}. Find State the expression that relates [𝑣]𝐡 with [𝑣]𝐢 and 𝑃 . 𝐢←𝐡 𝑃 . 𝐢←𝐡 Problem 4: 5 (A) i) ii) iii) iv) Let 𝐡 = {[1 0],[0 1],[0 0],[0 0]} be a basis for 𝑀22(𝑹) and 𝐢 = {1,π‘₯,π‘₯2,π‘₯3} 00001001 be a basis for 𝑃 (𝑹). Let T be a linear transformation given by: 3 𝑇(π‘Ž 𝑏)=π‘Žπ‘₯3 +(π‘βˆ’π‘+𝑑)π‘₯2 +(𝑐+2𝑑)π‘₯+(π‘‘βˆ’π‘Ž) 𝑐𝑑 Find the matrix representation of transformation 𝑇 with respect to basis 𝐡 β†’ 𝐢. Use part i) to find the matrix representation of transformation π‘‡βˆ’1 from 𝐢 β†’ 𝐡. Use part ii) to find the inverse transformation π‘‡βˆ’1(π‘Ž + 𝑏π‘₯ + 𝑐π‘₯2 + 𝑑π‘₯3) Is 𝑇 injective, surjective, both, or neither. How do you know? Problem 6: (A) i) ii) 011 Consider a basis 𝐡 = {[ 0 ],[ 0 ],[βˆ’1]} for the space π‘†π‘π‘Žπ‘›{𝐡}. βˆ’1 βˆ’1 3 0 1 βˆ’1 Use the Gram Schmidt Process on 𝐡 to find an orthonormal basis for π‘†π‘π‘Žπ‘›{𝐡}. Let matrix 𝐴 be created using the vectors in 𝐡 as the columns of 𝐴. Find the 𝑄𝑅 factorization for matrix 𝐴. 7

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Linear Algebra Exercises
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