[SOLVED] ECE269 Homework 2

1. Problem 1: Affine functions. A function f : Rⁿ → R^m is said to be affine if for any x, y ∈Rⁿ and any α, β∈R with α + β = 1, we have

f(αx + βy) = αf(x) + βf(y).

Note that without the restriction α + β = 1, this would be the definition of linearity.

• Suppose that A ∈R^m×n and b ∈R^m. Show that the function f(x) = Ax + b is affine.
• Prove the converse, namely, show that any affine function f can be represented uniquely as f(x) = Ax + b for some A ∈R^m×n and b ∈R^m.

2. Problem 2: Linear Maps and Differentiation of polynomials. Let P_n be the vector space consisting of all polynomials of degree ≤ n with real coefficients. (a) Consider the transformation T : P_n →P_n defined by

dp(x)

T(p(x)) =   .

dx

For example, T(1 + 3x + x²) = 3 + 2x. Show that T is linear.

(b) Using {1, x, . . . , xⁿ} as a basis, represent the transformation in part (a) by a matrix A ∈R(n+1)×(n+1). Find the rank of A.

3. Problem 3: Matrix Rank Inequalities.

Show the following identities about rank. (a) If A ∈F^m×n,B ∈F^n×k then

rank(B) ≤ rank(AB) + dim(null(A))

• If A ∈F^m×n,B ∈F^m×n then

rank(A + B) ≤ rank(A) + rank(B)

• Suppose A, B ∈F^m×m. Then show that if AB = 0 then

rank(A) + rank(B) ≤ m

• Suppose A ∈F^m×m. Then show A² = A if and only if

rank(A) + rank(A − I) = m

where I ∈F^m×m is the identity matrix.

4. Problem 4: Solution of Linear System of Equations. Consider the system of linear equations

y = ABx

where A ∈R^n×n and B ∈R^n×m, m ≤ n. For each of the following cases, find conditions (in terms of null spaces and range spaces of A and B) under which there can be a unique solution, no solution, or infinite number of solutions.

• rank(A) = n, and rank(B) = m.
• rank(A) = n, and rank(B) < m. (c) rank(A) < n, and rank(B) = m.

5. Problem 5: Infinite Dimensional Vector Spaces. Recall that C⁰([0,1]) is defined as the set of all continuous functions f : [0,1] → R, is a vector space over R. Let S = {1, (x + 1), (x + 2)², (x + 3)³, . . . , (x + i)ⁱ, . . . }.
   • Is there a vector in S which can be represented as a finite linear combination of other vectors in S?
   • Can any vector in C⁰([0,1]) be represented as a finite linear combination of vectors in S?

[Finite linear combination is a linear combination with finite number of terms]