Published on 31 Jan 2019

School

Department

Course

Professor

MT Q S

İ S. Y

A ,

Surname, Name:

Q . §. C S

Deﬁne T:M2×2ÏM2×2by T(A) = A−At.

(a) Show that Tis a linear transformation.

(b) Find a basis for the range of T

(c) Find a basis for the kernel of T.

(d) Verify that the Rank Theorem holds.

A

(a) Note that

T(A+B) = (A+B)−(A+B)t

=A+B−At−Bt

=A−At+B−Bt

=T(A) + T(B).

and

T(cA) = (cA)−(cA)t

=cA −cAt

=c(A−At)

=cT(A).

Thus, Tis linear.

(b) Consider M2×2with the standard basis: {[ 1 0

0 0 ],[0 1

0 0 ],[0 0

1 0 ],[0 0

0 1 ]}.

Let A=[a b

c d ]ÑA−At=[0b−c

c−b0]. Then we get T′:R4ÏR4

so that T′

a

b

c

d

=

0

b−c

c−b

0

where the standard matrix of T′is

0 0 0 0

0 1 −1 0

0−1 1 0

0 0 0 0

∼

0 0 0 0

0 1 −1 0

0 0 0 0

0 0 0 0

.

1

## Document Summary

T(a + b) = (a + b) (a + b)t. = a at + b bt. Thus, t is linear. (b) consider m2 2 with the standard basis: {[ 1 0. Let a = [ a b c d ] a at = [ so that t . 0 b c a b c d. Then we get t : r4 r4. 0 1 ]}. c b where the standard matrix of t is. The range of t is given as x2 which implies that. : x2 r range of t = {x2[ 1 0 ] : x2 r} . 1 0 ]} is a basis for the range of t. Thus, the set {[ (c) note that the null space of t is null t = implies that kernel of t = {x1[ 1 0. 0 1 ] : x1, x2, x4 r} .