7/27/2019 Linear Algebra - Exercise 3

1/2

10B1NMA731 APPLIED LINEAR ALGEBRA

EXERCISE-3 Linear Transformation

1. Define a linear transformation from one vector space to another. For a linear transformation T

show that 0)0( =T . If 0)( =uT can we say that 0=u ?

2. Determine whether following are linear transformations

(a) 32: RRL defined by ),,(),( yxxyyxL +=

(b) 23: RRL defined by ),(),,( yzxyzyxT =

(c) 22: RRL defined by ),(),( 22 xyyxyxf +=

(d) RRL 2: defined by yxyxf =),(

3. Let 23: RRT be a transformation defined by ).,(),,( zyxzyxzyxT +++= Show that it

is linear. Will it still be linear if zyx ++ is replaced by 1+++ zyx . Find the subset in 3R

which is mapped into 0 of 2R . Is it a subspace of 3R ?

4. Let22

2:

RPT be the transformation defined by

=+ ab

babxaT )( . Is T linear? Is the image

set a subspace of 22R ? If yes, find its basis and dimension.

5. Show that the set of all linear transformations from V to W with suitable definitions of

addition and scalar multiplication is a vector space over the same field.

6. Define the terms null space and range space of a linear transformation. Prove that both are

subspaces. Show that for a one to one linear transformation, the null space comprises only the

zero element. Find null space and the range space of the linear transformation defined in Q. 3.Also find a basis and dimension for both.

7. Let WVT : be a linear transformation and let )(TN and )(TR be the null space and the

range space respectively. Define the terms rank and nullity of T and show that rank +)(T

nullity =)(T dimV . Verify this theorem for the linear transformations of Q.3 and 4.

8. Show that a linear transformation T from a vector space V (dimension n ) to a vector spaceW (dimension m ) can be represented by an nm matrix. Does the matrix depend on the

bases in V and W ? Let 32: RRT be defined by )32,,(),( yxyxyxyxT ++= . Using

the standard bases viz )}1,0(),0,1{(1 =B in2R , and )}1,0,0(),0,1,0(),0,0,1{(2 =B in

3R , find the matrix representation of T . Do the same exercise when the bases are

)}1,1(),0,1{(1 =C in 2R and )}0,0,1(),0,1,1(),1,1,1{(2 =C in 3R .

7/27/2019 Linear Algebra - Exercise 3

2/2

9. Let 33: RRT be the linear transformation defined by

=

/

/

/

511

131

211

z

y

x

z

y

x

. Find )(),( TRTN

and their dimensions and verify the rank and nullity theorem. (Note that the rank of the

coefficient matrix is the same as rank ofT )