GATE MA topic:linear-algebra PYQs & Answers — Page 6

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topic:linear-algebra

Questions 51–60 of 62. Imported answers require independent review unless marked reviewed.

Topic practice 62 questionsPage 6 of 7

Consider the inner product space

of all polynomials of degree at most 2 over the field of real numbers with the inner product

for

. Let

be an orthogonal set in

, where

and

are real constants. Then the value of

is equal to _____.

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Let

, where

are real numbers and

. The geometric multiplicity of the largest eigenvalue of

equals ________.

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Consider the subspaces

. Then the dimension of

equals ________.

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Let

be the real vector space of all polynomials of degree less than or equal to 2 with real coefficients. Let

be the linear transformation given by

where

is the derivative of

. Then the number of nonzero entries in the Jordan canonical form of a matrix of

equals ________.

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Let

be the vector space of all

real matrices over the field

. Define the linear transformation

, where

denotes the transpose of the matrix

. Then the trace of

equals _________.

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Consider

with the usual inner product. If

is the distance from

to the subspace

, then

_________.

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Consider the matrix

with

, where

is the

identity matrix and

is the transpose of

. If

and

are two distinct eigenvalues of

, then

_________.

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Consider the vector space

of polynomials of degree at most 2. Let

be a linear functional such that

and

. Then

equals ______.

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Let

be a

matrix such that

, where

is the identity matrix. If

has two distinct eigenvalues and each eigenvalue has geometric multiplicity 3, then the total number of nonzero entries in the Jordan canonical form of

equals ______.

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and

are the null spaces of

and

, respectively, then the dimension of the subspace

equals ________.

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GATE MA topic:linear-algebra PYQs

Question 53

Question 13

Question 14

Question 15

Question 40

Question 41

Question 42

Question 1

Question 2

Question 26