Optionals IAS Mains Mathematics: Questions 80 - 86 of 283

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Question 80

Matrix of a Linear Transformation

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

Matrix of a Linear Transformation

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

Inverse of a Matrix

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

Linear Transformations

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

Linear Transformations

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

Linear Transformations

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.