Describe various spaces associated with an m × n matrix A, such
as null space, row space. column space and eigenspace. What are the
relationships among them? How does the concept of a linear
transformation and its properties relate to matrices and those
spaces of the matrices?
Let A be an m × n matrix and B be an m × p matrix. Let C =[A |
B] be an m×(n + p) matrix.
(a) Show that R(C) = R(A) + R(B), where R(·) denotes the range
of a matrix.
(b) Show that rank(C) = rank(A) + rank(B)−dim(R(A)∩R(B)).
1. For an m x n matrix A, the Column Space of A is a subspace of
what vector space?
2. For an m x n matrix A, the Null Space of A is a subspace of
what vector space?
Suppose C is a m × n matrix and A is a n × m matrix. Assume CA =
Im (Im is the m × m identity matrix). Consider the n × m system Ax
= b.
1. Show that if this system is consistent then the solution is
unique.
2. If C = [0 ?5 1
3 0 ?1]
and A = [2 ?3
1 ?2
6 10] ,,
find x (if it exists) when
(a) b =[1...
2. Using matrices, create an algorithm that takes a matrix of
dimension N x N and feed it in a spiral shape with the sequential
number from 1 to N^2.
Then do an algorithm in PSEint
In an attempt to explain what an elder matrix is: it is a n by m
matrix in which each row and each column is sorted in ascending
order.
Inputs in the matrix can either be finite integers or ∞. the
∞symbol is used when accounting for nonexistent inputs.
for all questions below please answer using pseudocode and
explainations
(a) create an algorithm EXTRACT-MIN on an Elder Matrix that is not
empty. the algorithm must run in O(m+n) time. The...
Prove via induction the following properties of Pascal’s
Triangle:
•P(n,2)=(n(n-1))/2
• P(n+m+1,n) = P(n+m,n)+P(n+m−1,n−1)+P(n+m−2,n−2)+···+P(m,0) for
all m ≥ 0
Let A be a diagonalizable n × n matrix and let P be an
invertible n × n matrix such that B = P−1AP is the diagonal form of
A. Prove that Ak = PBkP−1, where k is a positive integer. Use the
result above to find the indicated power of A. A = 6 0 −4 7 −1 −4 6
0 −4 , A5 A5 =
Let A be a diagonalizable n × n
matrix and let P be an invertible n × n
matrix such that
B = P−1AP
is the diagonal form of A. Prove that
Ak = PBkP−1,
where k is a positive integer.
Use the result above to find A5
A =
4
0
−4
5
−1
−4
6
0
−6