# Optionals IAS Mains Mathematics: Questions 80 - 86 of 283

Access detailed explanations (illustrated with images and videos) to **283** questions. Access all new questions- tracking exam pattern and syllabus. View the complete topic-wise distribution of questions. *Unlimited Access, Unlimited Time, on Unlimited Devices*!

View Sample Explanation or View Features.

Rs. 750.00 -OR-

How to register? Already Subscribed?

## Question 80

Appeared in Year: *2008*

### Describe in Detail

Essay▾Find the dimension of the subspace of spanned by the set

Hence find a basis for the subspace. (CS main Paper 1)

### Explanation

Let .

Let be the subspace of spanned by where

Let us construct a Matrix

Whose rows are the given vectors of and convert it into the echelon from.

Clearly which is in echelon from and the number of non-zero rows are equal to 3. Corresponding these rows the vectors of .

from a basis of .

i.e.. is a Maximum number of linearly independent subset of a…

… (4 more words) …

## Question 81

Appeared in Year: *2013*

### Describe in Detail

Essay▾Show the vectors and in are linearly independent over the field of real numbers but are linearly dependent over the complex numbers. (CS Paper I)

### Explanation

Let when are scalar.

It is can be written as

Operate we get

Operate we get

Operate we get

and

and .

If

If . Then Let

Which is not true

So, are L. I over

If

So,

So, are L. D over .

## Question 82

Appeared in Year: *2008*

### Describe in Detail

Essay▾Let be a non-singular Matrix show that if then . (CS main Paper 1)

### Explanation

Given that is Non-singular Metrix

exists.

And Premultiply by on both sides, we get

Substituting this into (1) , we get

Which is the required Result.

## Question 83

Appeared in Year: *2014*

### Describe in Detail

Essay▾Prove that the function where

Satisfies Cauchy-Riemann equations at the origin, but the derivative of at does not exist. (CS main Paper 2)

### Explanation

Here, , (where

Now,

and

Cauchy-Riemann equations are satisfied at .

Now,

Let along then we have

Further, let along , then we have

is not unique.

Thus does not exist at the origin.

## Question 84

Appeared in Year: *2012*

### Describe in Detail

Essay▾Consider the linear mapping by

Find the matrix relative to the basis and the matrix relative to the basis { (1,2) , (2,3) } . (CS main Paper 1)

### Explanation

Given by

And

Now,

And

Now, the Matrix of w. r. t bases is

Now, we find Matrix of w. r. t basis

Matrix of w. r. t Basis is

## Question 85

Appeared in Year: *2012*

### Describe in Detail

Essay▾If is a characteristic root of a non-singular matrix , then prove that is a characteristic root of . (CS main Paper 1)

### Explanation

Since is an eigen value of a non-singular matrix

and there exist a non-zero column vector such that

is eigen values of .

## Question 86

Appeared in Year: *2009*

### Describe in Detail

Essay▾Prove that the set of the vectors in which satisfy the equations and is a subspace of . What is the dimension of this subspace find one of its bases. (CS main Paper 1)

### Explanation

Let be the given vector space.

Let

Since

is Non-empty subset of .

Let

Then

Let , then we have

Since

And

Now, we have

Let

Then where

Since

From

i.e.. spans the subspace of

Since no vector of is a Scalar multiple

is basis of

Since number of elements in basis S is 2.