Matrices and Vector Spaces 1

Consider the equation 2x+2y=1, 2x-y=1 over z₃. What is the solution of (x,y)

• (1,1) but not (2,0)

• (2,0) but not (1,1)

• Both (1,1) and (2,0)

• (1/2,0)

What is the dimension of the vector space formed by the solution of the system of the following equations?

x₁ +x₂ + x₃ =0

x₁ +2x₂ =0

x₂ -x₃ =0

• 1

• 2

• 3

• 0

The dimension of the subspace of R³ spanned by (-3,0,1), (1,2,1),(3,0,-1)

• 0

• 1

• 2

• 3

Consider the following assertions.

I. Rank (ST) = Rank (S) =Rank (T)

II. Rank (ST) = Rank(S), if T is non singular

III. Rank (ST) = Rank (T) , if T is non singular

where S,T:V -V are linear transformations of a finite dimensional vector space V. Which of these is/are correct?

• Only I

• Only II

• I and II

• II and III

The system of equation kx+y+z=1, x+ky+z=k, and x+y+kz=k³ does not have a solution, if k is equal to

• 0

• 1

• -1

• -2

Which one of the following statement is correct?

• There is no vector space of dimension 1

• Any three vectors of a vector space of dimension 3 are inearly independent

• There is one and only one basis of a vector space of finite dimension

• If a non zero vector space V is generated by a finite set S, then V can be generated by a linearly independent subset of S

Consider the following statement:

Assertion(A): If M is a nx n matrix with rank (n-1), then M can be made non singular by changing one element.

Reason (R): An nxn matrix with rank n-1 , has a non vanishing minor of order n-1 of these statement.

• Both A and R are true and R is the correct explanation of A

• Both A and R are true, but R is not a correct explanation of A

• A is true but R is false

• A is false but R is true.

The system of equation

2x+y=5

x-3y=-1

3x+4y=k

is consistent, when k is

• 1

• 2

• 5

• 10

Let A be a mxn matrix with row rank= r= column rank. The dimension of the space of solutions of the system of linear equation AX=0 is

• r

• n-r

• m-r

• min(m,n)-r

Let A and B be nxn matrices with the same minimal polynomial. Then

• A is similar to B

• A is diagonalizable, if B is diagonalizable

• A-B is singular

• A and B commute