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QUESTION: 1

The system of equations

Solution:

We are given that the system of equation,

It may be written in the matrix form as,

Here, the coefficient of matrix is,

and the augmented matrix is given by,

Reduce the system equation in echelon form using the operations

R_{1}" and

R_{3} -> R_{3} - R_{1}

These operation yield:

and also, R_{3} --> R_{3} - 2R_{2
}

Here, Rank of A = Rank of aug(A) < number of unknowns.

Hence, the given system of equation has infinite many solutions.

QUESTION: 2

Which one of the following is true?

Solution:

We need to find the dim Horn (P_{3}t, R^{2})

Since dim (P_{3}t) = 3

and dim R^{2} = 2

Therefore,dim Hom (P_{3}t), R^{2})

= 3 x 2 = 6

QUESTION: 3

Which one of the following is true?

Solution:

We need to find the dimension of Horn (M_{2,}_{3}, R^{4}).

Since dim M_{2,}_{3} = 2 x 3 = 6

and dim R^{4} = 4

Therefore,dim Horn (M_{2,}_{3}, R^{4})

= 6 x 4 = 24

Let M_{n,m} be the vector space of all n x m matrices over R. Then

dim M_{n,m} = n x m

QUESTION: 4

Let :R^{3} --> R^{2} be the linear transformation given by

T(x, y, z) = (x, y),

with respect to standard basis of R^{3} and the basis {(1,0), (1, 1)} of R^{3}. What is the matrix representation of T?

Solution:

Let T : R^{3}--> R^{2} be the linear transformation defined by

T(x, y, z) = (x, y)

we need to determine the matrix of linear transformation T w.r.t. standard basis of R^{3} and the basis {(1, 0), (1 ,1 } of R^{2}. since {(1 ,0 ,0) , (0,1,0) (0,0,1)} be the standard basis of R^{3}, therefore,

Thus, the matrix of linear transformation T w.r.t. the standard basis R^{3} and the basis

QUESTION: 5

Let T : R^{3} --> R^{3} be defined by T(x, y, z) = (x, y, 0) and S : R^{2} —> R^{2} be defined by S(x, y) = (2x, 3y), be linear transformations on the real vector spaces R^{3} and R^{2}, respectively. Then, which one of the following statement is correct?

Solution:

Let T : R^{3} -> R^{3} be the linear transformation defined by

and S : R^{2} ---> R^{2} be the linear transformation defined by

S(x, y) = (2x, 3y)

Now,ker T = {(x, y , z) such that T(x,y, z) = 0}

= {(x, y ,z) : (x, y, 0) = (0,0,0)}

= {(0,0,z) : z ∈ R }

therefore, ker T ≠ {0, 0, 0} hence T is singular.

Next,ker S = {(x, y ) : S(x, y) = 0}

= {(x,y) : (2x,3y) = (0, 0)}

= {(0, 0)}

Therefore, S is non-singular.

QUESTION: 6

Consider the equation AX = B, where A = and B = , then

Solution:

We are given that the equation AX = B, where

Hence, the rank of A = 2 and, the augmented matrix is given by,

The rank of A = rank of aug (A) = 2

= number of unknowns.

Hence, there exist a unique solutions.

QUESTION: 7

The system of equation 2x + y = 5, x - 3y = -1

3x + 4y = k is consistent, when k is

Solution:

We are given that the system of equations,

2x + y = 5

x - 3y = - 1

3x + 4y = k is consistent.

We need to find the value of k. The given system of equation may be written as,

Since, this system of equation is consistent. Therefore

or 2(-3k + 4) -1(k +3) + 5(4 + 9) = 0

-7k + 70 = 0

k =10

QUESTION: 8

The value of α for which the system of equations

x + y + z = 0

y + 2z = 0

αx + z = 0 has more than one solution is

Solution:

We are given that the system of equation,

x + y + z =0

y + 2z = 0

αx + z= 0

has more than one solution. We need to find the value of α. The determinant of the coefficient matrix must be zero, i.e.,

QUESTION: 9

The system of the equations:

x + 2y + z - 9

2x + y + 3z=7

can be expressed as

Solution:

We are given that the system of equations,

x + 2y + z = 9

2x + y + 3z = 7

This system of equation can be expressed in the form AX = B i.e.

which is required matrix form.

QUESTION: 10

Let A be an n x n matrix from the set of numbers and A^{3} - 3A^{2} + 4A - 6I = 0, w hereI is nxn unit matrix. If A^{-1 }exist, then

Solution:

We are given that A be an n x n matrix from the set of real num bers and A^{3} - 3A^{2} + 4A - 6I = 0 ,

where, I is n x n unit matrix.

Since,

A^{3} - 3A^{2} + 4A = 6I

QUESTION: 11

If T : R^{2} --> R^{3} is a linear transformation T(1, 0) = (2, 3, 1) and T(1,1) = (3,0,2), then which one of the following is correct?

Solution:

We are given that a linear transformation T: R^{2}--> R^{3}

show that

T(1,0) = (2,3,1)

and T(1 , 1) = (3, 0, 2)

We need to determine the image of (x, y) under the linear transformation T.

Let there exist scalars α, β,

such that (x,y) = a(l,0) + β(l, 1) or equivalently

(x,y) =(α + β, β)

Comparing the components of the co-ordinates we get,

α + β = x, β = y

Solving for α and β, we get

α = x-y, β = y

therefore, (x, y) = (x - y) (1,0) + y( 1,1) taking the image under linear transformation T, we get

Using the linearity condition, we get T(x,y) = (x-y)T(l,0)+yT(l,l)

Substituting the value of T(1,0) and T(1, 1) we get

T(x, y) = (x -y) (2, 3, 1) + y(3, 0, 2)

= (2x -2y + 3y, 3x - 3y, x - y + 2y)

- (2x + y, 3x - 3y, x + y)

Therefore , the image of (x, y) under T that is

QUESTION: 12

Consider, the linear transformation

T : R^{4 }----> R^{4} given by:

T(x, y, z, u) = (x, y, 0, 0),

Then, which one of the following is correct?

Solution:

We are given that linear transformation

We need to determine Rank and Nullity of T.

Let (x,y, z, u) ∈ ker T

Then T(x, y, z, u) = (0, 0, 0, 0)

Using the definition of linear transformation we get

(x,y, 0, 0) = (0, 0, 0, 0)

implies x = 0, y = 0, z and u are arbitrary

Therefore,

Hence, Nullity of T = 2

Using Rank Nullity theorem, we get

QUESTION: 13

What is the rank of the linear transformation T : R^{3} ---> R^{3} defined by T(x, y, z) = (y, 0, z)?

Solution:

We are given that the linear transformation T : R^{3} ---> R^{3} defined by

We need to determine the rank of the linear transformation T.

Let (x, y, z) ∈ ker T

ThenT(x, y, z) = (0, 0, 0)

Using the definition of linear transformation, we get,

implies y = 0, z = 0 and x is arbitrary Therefore, ker T = {( x, y, z ) : y = 0, z = 0 and x is arbitrary}

Hence, Nullify of T= 1

Using Rank Nullity theorem, we get Rank T= dim R^{3} - Nullity T

= 3 - 1 = 2

QUESTION: 14

Let V be the vector space of all 2 x 2 matrix over the field R of real numbers and B = . If T : V--> V is a linear transformation defined by T(A) = AB - BA, then what is the dimension of the Kernel of T?

Solution:

Let V be the vector space of all 2 x 2 matrices over the field R of real numbers and matrix

Let T : V —> V be a linear transformation defined by T(A) = AB - BA

We need to determine the dim of ker T.

Let

Then, take T(A) = 0 implies AB - BA = 0

Substituting the values of matrices A and B,

we get

or equivalently

=

or equivalently

implies c =0 and a + b - d = 0 Therefore,

Hence, dim (ker T) = Total number of variables - Number of restrictions = 4 - 2 = 2

QUESTION: 15

Let T :R^{3 }---> R^{3} be a linear transformation given by T(x, y, z) . What is the rank of T?

Solution:

We are given that the linear transformation T : R^{3}---> R^{3 }defined by

We need to determine Rank of T.

Now,

Therefore, dim of ker T = 1

Hence by Rank Nullity theorem, we get Rank T = dim R^{3} - Nullity T

= 3 - 1 = 2

QUESTION: 16

Let T : R^{3} ---> R^{3} be a linear transformation given by T(x, y, z) = (x, y, 0). Then, the null space is generated by which one of the following?

Solution:

We are given that a linear transformation T : R^{3} —> R^{3} defined by

Therefore

(0,0,1) ∈ ker T and Hence, the Null space is generated by (0, 0,1).

QUESTION: 17

Consider the vector space C over R and let 7: C --> C be a linear transformation given by T(z) = z. Then, which one of the following is correct?

Solution:

We are given that a linear transformation T : C -->C defined by T(z) =

Where C is a vector space over the field of Real Number R and z is the conjugate of complex number z.

ker T= {z : T(z) = 0}

= { z : z = 0} = {0}

Hence, T is one-one.

Since C over R is a finite dimensional vector space (two dimensional vector space) and T : C --> C is one linear transformation. Therefore T is onto.

QUESTION: 18

Let T -->R^{2} ---> R be defined by T(x,y) = (x + y , x - y)

Which one of the following is the matrix of T for (0, 1) and (1, 0) as basis for both domain and range of T?

Solution:

We are given that the linear transformation T : R^{2} ---> R^{3} defined by

T(x,y) = ( x + y, x -y )

We need to determine the matrix of T with respect to basis {(0,1), (1, 0)}

Now

Hence, the matrix of linear transformation T with respect to the basis .

QUESTION: 19

Consider the mapping

Q. Which of the above are linear transformation?

Solution:

I. We are given that the transformation T: R^{3} —> R^{2} defined by

T (x , y , z) = (x + 1, y + z)

We need to determine the linearity of the given linear transformation.

Now,T'(0, 0, 0) = (1, 0)

Since the image of (0, 0,0) under transformation T is (1, 0) which is not the zero of R^{2}.

Hence, T is non-linear.

II. We are given that the transformation T : R^{3} --> R defined by

T(x,y,z) = (xy)

We need to determine the linearity of this given transformation.

Since the image of (x, y, z) under linear transformation.

T is an algebraic term of degree 2.

Therefore, T is non linear.

III. We are given that the transformation T: R^{3} —> R^{2} defined by

T(x, y, z) = (| x |, 0)

We need to determine the linearity of this linear transformation

Let (1, 0, 0) and (-1, 0, 0) be two vectors of R^{3}

Then T(l,0,0) = (l,0)

andT(-l, 0, 0) = (1, 0)

Now,T [(1, 0, 0) + (-1, 0, 0)] =

T(0, 0, 0) = (0, 0)

and T(1, 0, 0) + T(-1, 0, 0)

= (1, 0) + (1,0) = (2, 0)

Hence, T [(1, 0, 0) + (-1, 0, 0)] ≠ T{ 1, 0, 0) + T (-1, 0, 0).

Therefore, T is non linear.

QUESTION: 20

Let T : R^{2} --->R^{2} be a linear transformation such that T

What is the value of ?

Solution:

Let T : R^{2} —> R^{2} be a linear transformation such that

We need to determine the image of under linear tansformation T.

Let there exist scalars α and β such that

Since

and taking the image under T, we get

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