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