MATH661

1.Linear mappings

Exercise 1. Construct the matrix $𝑨$ that represents the mapping $𝒇 : ℝ^{3} \to ℝ^{3}$ , $𝒇$ reflects a vector across the $x_{1} x_{2}$ plane.

Exercise 2. Construct the matrix $𝑩$ that represents the mapping $𝒇 : ℝ^{3} \to ℝ^{3}$ , $𝒇$ reflects a vector across the $x_{2} x_{3}$ plane.

Exercise 3. Construct the matrix $𝑪$ that represents the mapping $𝒇 : ℝ^{3} \to ℝ^{3}$ , $𝒇$ reflects a vector across $x_{2}$ axis.

Exercise 4. Does $𝑪 = 𝑩 𝑨$ ? If so, what does this signify?

Exercise 5. Construct the matrix $𝑫$ that represents the mapping $𝒇 : ℝ^{3} \to ℝ^{3}$ , $𝒇$ rotates a vector around the $x_{3}$ axis by angle $θ$ .

Exercise 6. Construct the matrix $𝑬$ that represents the mapping $𝒇 : ℝ^{3} \to ℝ^{3}$ , $𝒇$ rotates a vector around the $x_{2}$ axis by angle $φ$ .

2.Norms

Exercise 11. For $𝒙 \in ℝ^{m}$ , prove ${|| 𝒙 ||}_{2} ⩽ \sqrt{m} {|| 𝒙 ||}_{\infty}$ .

Exercise 12. Prove that equality holds in ${|| 𝒙 + 𝒚 ||}_{2} ⩽ {|| 𝒙 ||}_{2} + {|| 𝒚 ||}_{2}$ if and only if $\exists a < ℝ$ , s.t. $𝒙 = a 𝒚$ .

Linear Algebra Tools - Exercises

1.Linear mappings

2.Norms