Linear Algebra Exercises

Problem 1: (A) Show𝑆 = {[ 𝑎 𝑎 ]∶𝑎+𝑏+𝑐=0, 𝑎,𝑏,𝑐,𝑑∈𝑹}isasubspaceof𝑀22(𝑹) 𝑏+𝑑 𝑐−𝑑 (B) i) Findthebasisanddimensionfor𝑆={[ 𝑎 𝑎 ]∶𝑎+𝑏+𝑐=0, 𝑎,𝑏,𝑐,𝑑∈𝑹} 𝑏+𝑑 𝑐−𝑑 ii) Explain why 𝑆𝑝𝑎𝑛{𝑆} ≠ 𝑀22(𝑹) 2 Problem 2: 3 (A) (B) (C) i) ii) Let 𝐵 = {1 + 𝑥 + 𝑥2, 1 − 𝑥 − 𝑥2, 1 − 𝑥 + 𝑥2} be a basis for 𝑃 (𝑹). 2 1 Determine 𝑤 when [𝑤]𝐵 = [ 0 ] −2 Determine [𝑦]𝐵 when 𝑦 = 4 + 2𝑥. Let 𝐵 = {1 + 𝑥 + 𝑥2, 1 − 𝑥 − 𝑥2, 1 − 𝑥 + 𝑥2} and 𝐶 = {1 + 𝑥, 2 − 𝑥2, 𝑥2}. Find State the expression that relates [𝑣]𝐵 with [𝑣]𝐶 and 𝑃 . 𝐶←𝐵 𝑃 . 𝐶←𝐵 Problem 4: 5 (A) i) ii) iii) iv) Let 𝐵 = {[1 0],[0 1],[0 0],[0 0]} be a basis for 𝑀22(𝑹) and 𝐶 = {1,𝑥,𝑥2,𝑥3} 00001001 be a basis for 𝑃 (𝑹). Let T be a linear transformation given by: 3 𝑇(𝑎 𝑏)=𝑎𝑥3 +(𝑏−𝑐+𝑑)𝑥2 +(𝑐+2𝑑)𝑥+(𝑑−𝑎) 𝑐𝑑 Find the matrix representation of transformation 𝑇 with respect to basis 𝐵 → 𝐶. Use part i) to find the matrix representation of transformation 𝑇−1 from 𝐶 → 𝐵. Use part ii) to find the inverse transformation 𝑇−1(𝑎 + 𝑏𝑥 + 𝑐𝑥2 + 𝑑𝑥3) Is 𝑇 injective, surjective, both, or neither. How do you know? Problem 6: (A) i) ii) 011 Consider a basis 𝐵 = {[ 0 ],[ 0 ],[−1]} for the space 𝑆𝑝𝑎𝑛{𝐵}. −1 −1 3 0 1 −1 Use the Gram Schmidt Process on 𝐵 to find an orthonormal basis for 𝑆𝑝𝑎𝑛{𝐵}. Let matrix 𝐴 be created using the vectors in 𝐵 as the columns of 𝐴. Find the 𝑄𝑅 factorization for matrix 𝐴. 7