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QUESTION: 1

If {v_{1}, v_{2}, v_{3}} is a linearly independent set of vectors in a vector space over which one of the following sets is also linearly independent?

Solution:

QUESTION: 2

Consdierthe subspace W = {(x_{1}, x_{2},.... ,x_{10}) x_{n} = x_{n-1} + x_{n - 2} for 3 __<__ n __<__ 10) of the vector space the dimension of W is

Solution:

QUESTION: 3

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

Solution:

QUESTION: 4

Which of the following subsets of a basis of

B_{1} = {(1,0,0,0), (1,1,0,0), (1,1,1,0), (1,1,1,))

B_{2} = {(1,0,0,0), (1,2,0,0), (1,2,3,0), (1,2,3,4)}

B_{3} = {(1,2,0,0), (0,0,1,1), (2,1,0,0), (-5,5,0,0))

Solution:

QUESTION: 5

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

Solution:

QUESTION: 6

Consider the following row vectors α_{1} = (1,1,0,1,0,0), α_{2} = (1,1,0,0,1,0), α_{3} = (1,1,0,0,0,1), α_{4} = (1,0,1,1,0,0), α_{5} = (1,0,1,0,1,0), α_{6} = (1,0,1,0,0,1)

The dimension of the vector space spanned by these row vectors is

Solution:

QUESTION: 7

Let the vector space of 10 x 10 matrices with entries in W_{A} be the subspace of spanned by [A^{n} : n __>__ 0} choose the correct statements.

Solution:

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

Solution:

QUESTION: 9

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

Solution:

QUESTION: 10

The dimension of the vector space of all symmetric matrices A = (a_{ij}) of order n x n (n __>__ 2) with real entries, a_{ii} = 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 H_{n} be the space of all n x n matrices A = (a_{ij}) with entries in satisfying a_{ij} = a_{rs} whenever i + j = r + s (i, j, r, s = 1, 2,..., n) then the dimention of H_{n}, as a vector space over

Solution:

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

Solution:

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?

Solution:

QUESTION: 14

Let V denote the vector space C^{5}[a, b] over R and

Solution:

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

Solution:

QUESTION: 16

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

Solution:

QUESTION: 17

The dimension of the vector space over the field

Solution:

QUESTION: 18

Then the dim V is

Solution:

QUESTION: 19

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

Solution:

QUESTION: 20

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

Solution:

The system of 3 equations in 4 unknowns is

1x + 1y +(-1)z +0w =0……..(1)

0x + 1y + 1z + 1w = 0……….(2)

2x + 1y +(-3)z +1w = 0……..(3). The row operation (3) - 2×(1) reduces (3) to

0x +(-1)y+(-1)z+(-1)w =0…….(4), which is just (-1)× (3). Hence the given system is equivalent to the pair of equations:

x = -y+z and y= -z -w. This pair is equivalent to x=(z+w)+z = 2z+w and y = -z -w, where z and w are completely arbitrary. Thus

[x,y,z,w] =[2z+w,-z-w,z,w] =

z[2,-1,1,0] + w[1,-1,0,1].

We had already observed that the rank of the echelon matrix equivalent to the given coefficient matrix is 2, and hence its nullity = 4–2 = 2, which is the same as the dimension of the solution space V. Thus a basis of V may be taken as

B={[2,-1,1,0], [1,-1,0,1]}.

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