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QUESTION: 1

Define T(v) = Av, where A = . Then T(v):

Solution:

We are given that

T(v) = Av

where A =

We need to define T(v)

Let v = , then T(v) = Av =

Which reflects v about the x_{2}-axis.

QUESTION: 2

If the linear transformation T(v) = Av rotates the vectors (-1, 0) and (0,1), x radians clockwise, then:

Solution:

We are given that the linear transformation T(v) = Av rotates the vectors (-1,0) and (0,1), π radians clockwise.

We need to find the matrix A.

We know that if a vector (a, b) is rotated through an angle a under T. Then

Therefore, the matrix of T is

A =

QUESTION: 3

If the linear transformation T(v) = Av rotates the vectors v_{1 }= (-1, 0) and v_{2} = (0, 1) clockwise π radians, the resulting vectors are:

Solution:

We are give that the linear transformation T(v) = Av rotates the vectors v_{1} = (-1,0) and v_{2} = (0,1) clockwise π radians. We need to find the image of v_{1 }= (-1, 0) and v_{2} = (0,1) under T

We know that if a vector (a, b) is rotated through an angle α under linear transformation T, then

Here v_{1} = (-1, 0) and a = π

Therefore,v_{2} = (0,1)

= (1,0)

and T(v_{2}) = T(0, 1)

= (0,-1)

QUESTION: 4

If the linear transformation T(v) = Av rotates the vectors v_{1} = (-1, 0) and v_{2} = (0, 1) clockwise π/2 radians, the resulting vectors are:

Solution:

We are given that the linear transformation T(v) = Av rotates the vector v_{1} = (-1, 0) and v_{2} = (0,1) clockwise π/2radians.

We know that if a vector (a, b) is rotated through an angle a clockwise direction under linear transformation T.Then

Here, v_{1 }= (-1 , 0) and a = π/2 radians

Therefore, T(v_{1}) = T(-1, 0)

= = (0,1)

Therefore, T(-1, 0) = (0,1)

Similarly, for v_{2} = (0, 1) and α = π/2,

and T(v_{2}) = T(0 ,1)

= (1,0)

Therefore, T(0,1) = (1 ,0 ).

QUESTION: 5

If the linear transformation T(v) = Av rotates the vectors (-1, 0) and (0, 1) clockwise π/2 radians then:

Solution:

We are given that the linear transformation T(v) = Av rotates the vectors (-1, 0) and (0, 1) clockwise π/2 radians. We need to find the matrix A.

We know that if a vector (a, b) is rotated through an angle a clockwise under T. Then

Here, (a, b) = (-1, 0) and a = π/2

Therefore, T (-1, 0)

= (0, 1) = 0(-1, 0) + 1(0, 1)

and (a, b) = (0,1), α = π/2

= (1, 0) = - 1(-1, 0) + 0(0, 1)

Therefore, the matrix of T is A =

QUESTION: 6

If A = satisfies the matrix equation A^{2} - kA + 2I = 0, then what is the value of k?

Solution:

We need to find the value o f k where the matrix

A = satisfies the equations A^{2} - kA + 2I = 0.

But A =

So,

Putting these values in the equation

A^{2} - kA + 2I= 0, we get

or

or 1 - 3k + 2 = 0 => k = 1

QUESTION: 7

Under which one of the following condition does the system of equations have a unique solution?

Solution:

We are given that the system of equations

Reduce this system of equations to echelon form, using the operations "R_{2} ----> + R_{2} - 2R_{1}" and R_{3} ---> R_{3 }- R_{1}.

These operations yields:

Also we have, this system has a unique solution so a - 8 ≠ 0

or a ≠ 8

QUESTION: 8

What is the determinant of the following matrix?

Solution:

We need to find the determinant of the given matrix,

determinant of A =

or det (A) =

det (A) = 1/n!

QUESTION: 9

If the system of equations

x - 2y - 3z = 1, (p + 2)z = 3, (2p + 1)y + z = 2 is inconsistent, then what is the value of P?

Solution:

We are given that the system of equations,

x - 2y - 3z = l, (p + 2)z =3, (2p + l) y + z = 2 is inconsistent. Then we need to find the value of p.

The augmented matrix of this system of equations is,

The solution of this matrix is inconsistent if

p + 2 = 0

or p = -2

QUESTION: 10

If x, y, z are in AP with common difference d and the rank of the matrix is 2, then the value of d and k are

Solution:

We are given that x, y, z are in A.P. with common difference d i. e.,

or

and the rank of is z.

We need to find the values of d and k. The given matrix can be written as,

Applying R_{2} -> R_{2}, R_{1} and R_{3 }--> R_{3 }-R_{2
}

Applying, R_{3} ---> R_{3} - R_{2},

QUESTION: 11

The linear transformation T(x, y) = can be written as:

Solution:

We have

or equivalently

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

QUESTION: 12

The linear transformation T(x, y) = (x + 2y, x - 2y), can be written as a matrix transformation T(x, y) where:

Solution:

We are given that a linear transformation defined by

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

Let B ={(1, 0), (0, 1)} be a standard basis then

T(1,0) = (1,1) = 1 (1 ,0)+ 1(0,1)

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

Therefore, the matrix of T with respect to the basis B is

QUESTION: 13

Is the transformation T(x, y, z) = (x, y, 0) linear?

Solution:

We are given that a transformation defined by

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

Let α and β be scalars and (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) be two vectors. Then

T[α(x_{1}, y_{1}, z_{1}) + β(x_{1}, y_{2}, z_{2})]

or equivalently

Hence, T is linear.

QUESTION: 14

What is the range of T(v) = Av where A = ?

Solution:

We are given that a linear transformation T defined by

T(v) = Av

where

Let Then

Let v_{1}, v_{2}, v_{3} ∈ ker T. Then

Using the definition of T, we get

or equivalently

or equivalently

v_{1} = v_{2 }= v_{3} = - v_{1}

Therefore,ker T= {(v_{1}, v_{1}, - v_{1}): v ∈ R}.

Therefore, dim ker T= 1

Using Rank-Nullity theorem|

Rank T= 3 - dim ker T

= 3 - 1 = 2

Therefore, the range of T is a plane in R^{3}

QUESTION: 15

Suppose T_{1}: V ---> U and T_{2 }: U —> W be two linear transformation, Then:

Solution:

QUESTION: 16

Which one of the following is true?

Solution:

We need to find the dim Horn (R^{3}, R^{4}).

Since dim R^{3} = 3 and dim R^{4} = 4

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

= 3 x 4 = 12

Let U and V be n and m dimensional vector space.

Then dim Hom (U, V) = nm

QUESTION: 17

If C is a non-singular matrix and B then

Solution:

We are given that C is a non-singular matrix and

B = or B is similar to The eigen values of are all zeroes.

Hence, B^{k} = 0

For k = 3

B^{3} = 0

QUESTION: 18

If X = , the rank of X^{T}X, where X^{T} denotes the transpose of X, is

Solution:

We are given that the matrix

X=

We need to find the rank of X^{T}X. The transpose of X,

X^{T}=

Now,

Rank (X^{T}X) = 3

QUESTION: 19

The system of equation kx +y + z = 1,x + ky + z = k and x + y + kz = k^{3} does not have a solution, if k is equal to

Solution:

We are given that the system of equation

does not have a solution. We need to find the value of k. The given linear system of equation can be written as,

This system of equation has no. solution if rank (A) # rank (aug A) the augmented matrix is,

This augmented matrix has no. solutions if k = -2.

QUESTION: 20

We are given that the system of equation

has

Solution:

We are given that the system of equation

It may be written as

Here, the rank of the given coefficient matrix ≤ 3 and number of unknowns are four.

Hence, rank < no. of unknowns this system of equations has infinitely many solutions.

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