Description
1. Problem 1: Affine functions. A function f : Rn → Rm 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) = α f(x) + β f(y).
Note that without the restriction α + β = 1, this would be the definition of linearity.
(a) Suppose that A ∈ Rm×n and b ∈ Rm. Show that the function f(x) = Ax + b is affine.
(b) Prove the converse, namely, show that any affine function f can be represented
uniquely as f(x) = Ax + b for some A ∈ Rm×n and b ∈ Rm.
2. Problem 2: Linear Maps and Differentiation of polynomials. Let Pn be the vector space
consisting of all polynomials of degree ≤ n with real coefficients.
(a) Consider the transformation T : Pn → Pn defined by
T(p(x)) = dp(x)
dx .
For example, T(1 + 3x + x
2
) = 3 + 2x. Show that T is linear.
(b) Using {1, x, . . . , x
n} as a basis, represent the transformation in part (a) by a matrix
A ∈ R(n+1)×(n+1)
. Find the rank of A.
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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))
(b) If A ∈ F
m×n
,B ∈ F
m×n
then
rank(A + B) ≤ rank(A) + rank(B)
(c) Suppose A, B ∈ F
m×m. Then show that if AB = 0 then
rank(A) + rank(B) ≤ m
(d) 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 ∈ Rn×n and B ∈ Rn×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.
(a) rank(A) = n, and rank(B) = m.
(b) rank(A) = n, and rank(B) < m.
(c) rank(A) < n, and rank(B) = m.
5. Problem 5: Infinite Dimensional Vector Spaces. Recall that C
0
([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
, . . . }.
(a) Is there a vector in S which can be represented as a finite linear combination of other
vectors in S?
(b) Can any vector in C
0
([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]
