Test: Linear Algebra - 2


20 Questions MCQ Test Topic-wise Tests & Solved Examples for IIT JAM Mathematics | Test: Linear Algebra - 2


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QUESTION: 1

If {v1, v2, v3} is a linearly independent set of vectors in a vector space over  which one of the following sets is also linearly independent?

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QUESTION: 2

Consdierthe subspace W = {(x1, x2,.... ,x10) xn = xn-1 + xn - 2 for 3 < n < 10) of the vector space  the dimension of W is

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QUESTION: 3

Which one of the following is a subspace of the vector space

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QUESTION: 4

Which of the following subsets of a basis of
B1 = {(1,0,0,0), (1,1,0,0), (1,1,1,0), (1,1,1,))
B2 = {(1,0,0,0), (1,2,0,0), (1,2,3,0), (1,2,3,4)}
B3 = {(1,2,0,0), (0,0,1,1), (2,1,0,0), (-5,5,0,0))

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QUESTION: 5

Which of the following sets of functions from is a vector space over

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QUESTION: 6

Consider the following row vectors α1 = (1,1,0,1,0,0), α2 = (1,1,0,0,1,0), α3 = (1,1,0,0,0,1), α4 = (1,0,1,1,0,0), α5 = (1,0,1,0,1,0), α6 = (1,0,1,0,0,1)
The dimension of the vector space spanned by these row vectors is 

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QUESTION: 7

Let  the vector space of 10 x 10 matrices with entries in WA be the subspace of spanned by [An : n > 0} choose the correct statements.

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QUESTION: 8

Let V be a 3 dimensional vector space over the field  of 3 elements. The number of distinct 1 dimensional subspaces of V is

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QUESTION: 9

Let M = {(a1,a2,a3): ai ∈ (1,2,3,4); a1 + a2 + a3 = 6} then the number of elements in M is

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QUESTION: 10

The dimension of the vector space of all symmetric matrices A = (aij) of order n x n (n > 2) with real entries, aii = 0 and trace zero is

Solution:

The dimension of the space of n×n symmetric matrices with diagonal equal to zero is . Now, in your case the diagonal is not zero but the sum of its elements is zero, that means that you have n−1 elements which can vary. SO you get as expected.

QUESTION: 11

Let n be a positive integer and let Hn be the space of all n x n matrices A = (aij) with entries in satisfying aij = ars whenever i + j = r + s (i, j, r, s = 1, 2,..., n) then the dimention of Hn, as a vector space over

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QUESTION: 12

The dimension of the vector space of all symmetric matrices of order n x n (n > 2) with real entries and trace equal to zero is

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QUESTION: 13

Let {X,Y,Z) be a basis of Consider the following statements P and Q
P : {X + Y,Y + Z, X - Z) is a basis of
Q : {X + Y + Z,X + 2Y - Z, X - 3Z} is a basis of

Which of the above statements hold true? 

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QUESTION: 14

Let V denote the vector space C5[a, b] over R and

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QUESTION: 15

Let M be the space of all 4 x 3 matrices with entries in the finite field of three elements. Then the number of matrices of Rank three In M is

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QUESTION: 16

Consider the subspace W = {[aij] : aij = 0, if i is even) of all 10 x 10 matrices(real) then the dimension of W is

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QUESTION: 17

The dimension of the vector space  over the field

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QUESTION: 18

 Then the dim V is

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QUESTION: 19

 and let V = {(x, y, z) : |A| = 0}.Then the dimension of V equals

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QUESTION: 20

A basis of V = {(x, y, z, w) : x + y - z = 0, y + z + w = 0, 2x + y - 3z = 0)

Solution:

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