EE221A Linear System Theory Problem Set 2 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2014 Issued 9/16; Due 9/25 All answers must be justified. Problem 1: Solutions to linear equations (this was part of Professor El Ghaoui’s prelim question last year). Consider the set S = {x : Ax = b} where A ∈ Rm×n , b ∈ Rm are given. What is the dimension of S? Does it depend on b? Problem 2: Linearity. Given A, B, C, X ∈ Cn×n , determine if the following maps (involving matrix multiplication) from Cn×n → Cn×n are linear. 1. X 7→ AX + XB 2. X 7→ AX + BXC 3. X 7→ AX + XBX Problem 3: Rank-Nullity Theorem. Let A be a linear map from U to V with dimU = n and dimV = m. Show that dimR(A) + dimN (A) = n Problem 4: Matrix Representation of a Linear Map. Let A : (F 2 , F ) → (F 3 , F ) be defined by A(x, y) = (x + 3y, 2x + 5y, 7x + 9y). What is the matrix representation of A with respect to the standard bases, and With respect to new bases: 0 0 1 1 1 }, BF 3 = { 1 , 1 , 1 }? , BF 2 = { 0 1 1 0 0 Problem 5: Matrix Representation of a Linear Map. Let A : (U, F ) → (V, F ) with dim U = n and dim V = m be a linear map with rank(A) = k. Show that there exist bases (ui )ni=1 , and (vj )m j=1 of U, V respectively such that with respect to these bases A is represented by the block diagonal matrix I 0 A= 0 0 What are the dimensions of the different blocks? Problem 6: Sylvester’s Inequality. In class, we’ve discussed the Range of a linear map, denoting the rank of the map as the dimension of its range. Since all linear maps between finite dimensional vector spaces can be represented as matrix multiplication, the rank of such a linear map is the same as the rank of its matrix representation. Given A ∈ Rm×n and B ∈ Rn×p show that rank(A) + rank(B) − n ≤ rank AB ≤ min [rank(A), rank(B)] Problem 7: Norms. Show that for x ∈ Rn , √1 ||x||1 n ≤ ||x||2 ≤ ||x||1 . Problem 8. Prove that the induced matrix norm: ||A||1,i = maxj∈{1,...,n} 1 Pm i=1 |aij |. Problem 9. Consider an inner product space V , with x, y ∈ V . Show, using properties of the inner product, that ||x + y||2 + ||x − y||2 = 2||x||2 + 2||y||2 where || · || is the norm induced by the inner product. Problem 10. Consider an inner product space (Cn , C), equipped with the standard inner product in Cn , and a map A : Cn → Cn which consists of matrix multiplication by an n × n matrix A. Find the adjoint of A. 2

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