Faculty of Mechanical and Aerospace Engineering
Institut Teknologi Bandung
WF2102 Linear Algebra and Differential Equations
HW#2, due date: Wednesday, 24 September 2025 at 15:00 in the classroom
1.
Determine whether the mapping T is a linear transformation,
𝑇: 𝑀22 → 𝑅, where
𝑎 𝑏
𝑇 ([
]) = 3𝑎 − 4𝑏 2 + 𝑐 − 𝑑
𝑐 𝑑
2.
Let 𝑇1 (𝑥1 , 𝑥2 ) = (𝑥1 + 𝑥2 , 𝑥1 − 𝑥2 ) and 𝑇2 (𝑥1 , 𝑥2 ) = (3𝑥1 , 2𝑥1 +4𝑥2 ).
a. Find the standard matrices for T1 and T2
b. Find the standard matrices for 𝑇2 ∘ 𝑇1 and 𝑇1 ∘ 𝑇2
c. Use the matrices obtained in part (b) to find formulas for 𝑇1 (𝑇2 (𝑥1 , 𝑥2 )) and 𝑇2 (𝑇1 (𝑥1 , 𝑥2 )).
3.
Let 𝑇: 𝑃2 → 𝑃2 be the contraction operator with factor k = 1/4.
a) Find 𝑇(1 + 4𝑥 + 8𝑥 2 )
b) Find the rank and nullity of 𝑇.
4.
Consider the basis S = {v1, v2} for R2, where v1 = (-2,1) and v2 = (1,3), and let 𝑇: 𝑅 2 → 𝑅 3 be the linear
transformation such that
𝑇(v1) = (-1, 2, 0) and 𝑇(v2) = (0, -3, 5)
Find a formula for 𝑇(𝑥1 , 𝑥2 ), and use that formula to find 𝑇(2, −3).
5.
Compute (𝑇3 ∘ 𝑇2 ∘ 𝑇1 )(𝑥, 𝑦).
a) 𝑇1 (𝑥, 𝑦) = (𝑥 + 𝑦, 𝑦, −𝑥),
b) 𝑇2 (𝑥, 𝑦, 𝑧) = (0, 𝑥 + 𝑦 + 𝑧, 3𝑦),
c) 𝑇3 (𝑥, 𝑦, 𝑧) = (3𝑥 + 2𝑦, 4𝑧 − 𝑥 − 3𝑦)
Good luck,
FF and FB