Columbia University Linear Algebra and Square Matrices Questions

question 1 has 5 true and false question

detailed description is in the attached file.

Linear Algebra (MTH-SHU 140) Quiz of week 6
2020/03/27
The Lecturer
Name
Instructions. Please write down your answers in details.
Let R denote the set of real numbers.
1. Is the following each statement true or false? If false, write F and give counter examples. If
true, write T and give brief explainations.
(1) Let An
m
be a matrix. Then rank(P A) = rank(A) for any invertible matrix Pn
(2) Let u; v be nonzero vectors in
Rn :
Then
rank(uv T )
=
n:
rank(v T u):
(3) Let An m be a matrix and Pn n an invertible matrix. Then column space of A is the
same as that of P A; i.e. Spanfcolumns of Ag = Spanfcolumns of P Ag:
(4) Let A; B be two matrices of the same sizes. Then dim N ul(A)+dim N ul(B)
B):
dim N ul(A+
(5) Supose that AB = C for three square matrices A; B; C: If the rows of C are linearly
independent, then the rows of B are linearly independent.
1
1
; u2 =
1
two sets of vectors in R2 :
2. Let U = fu1 =
1
cos
g and V = fv1 =
1
sin
; v2 =
sin
cos
g (for any
(1) Prove that U; V are bases for R2 :
(2) Find the transition matrix P from U to V; i.e. [x]V = P [x]U for any x 2 R2 :
2
2 R) be
Bn m Dn
0
Cm
rank(B) + rank(C):
3. Let A =
l
be a partitioned matrix for matrices B; C; D. Prove that rank(A) =
l
3
4. Let M2
be the vector space of all 2 2 matrices. De…ne a function T : M2
a b
T (A) = A + AT for any A =
2 M2 2 :
c d
2
2
! M2
(1) Show that T is linear.
(2) Find a basis of M2
2:
(2) Find a standard matrix AT of the linear map T and compute the rank of AT :
4
2
by