Linear Algebra Optimization MCQ with answers



Q3.Identify the vectors which form a linearly independent set?

a. v1 = (1, -2, 3),  and v2 = (2, -4, 6)

b. v1 = (1, -2, 3),  and v2 = (-1, 2, -3)

c. v1 = (1, 0, 1),  and v2 = (0, 1, 1)

d. v1 = (1, -2, 3),  and v2 = (5, -10, 15)

Ans: C

Q4.The zero vector in the vector space R^4 is

a. (0,0,0)

b. (0,0,0,0)

c. None of the above

d. (0,0)

Ans: B

Q5.Which of the following step is not involved in Gauss Elimination Method?

a. Elimination of unknowns

b. Reduction to an upper triangular system

c. Evaluation of cofactors

d. Finding unknowns by back substitution

Ans: C

Q6.The vector space R3 and its subspace H = {(s,t,0) : s and t are real numbers} have the same  dimension




Q7.Let A be an n x n invertible matrix. Which of the following statements is false:

a. The equation A X = 0 has infinitely many solutions

b. A is row equivalent to the n x n identity matrix

c. The equation A X = b has at least one solution for each b in Rn

d. Ais an invertible matrix

Ans: A

Q8.Which of the following is correct?

a. R is a vector space over Z

b. R is a vector space over C

c. None of the above

d. R is a vector space over N

Ans: C

Q9.Let V be a vector space and W be a subspace of V then

a. ku  W,  u  W, k is a scalar

b. All of the above

c. u + v = v + u , u, v  W

d. m(nu) = (mn)u ,  u  W, m & n are scalars

Ans: B

Q10.Read the statements and choose the correct option.    

Statement A : In the matrix notation of system of equations [A][x] = [b] if all b’s are zero then the system is called homogeneous system.                                   

Statement B : In the matrix notation of system of equations [A][x] = [b] if at least one of the b’s is not zero then the system is called non homogeneous system.

Show Answer

Ans: both A and B are true

Q11.The system of equations 4x+y-3z-w =0,2x+3y+z-5w=0,x-2y-2z+3w=0 has,

Ans: infinite number of solutions

Q12.for what value of A,do the simultaneous equation 2x+3y=5,4x+6y=λ have infinite solution?

Ans: λ = 10

Q13.if V1 and V2 are 3- dimensional subspaces of a 4 dimensional vector space V,then smallest possible dimension of V1 V2 is

Ans: 2

Q14.Consider the vector space V= R4 then the trivial subspaces of V are:

Ans: (0,0,0,0) AND R

Q15.If A &B are any two matrices,then which of the following is true

Ans: AB cannot necessarily be defined

Q16.Identify the vectors which form a linearly independent set?

Ans: v1=(1,0,1) and v2=(0,1,1)

Q17.If A and B are matrices of order n x n and m x n respectively, then which of the following are defined,

Ans : BA,A2

Q18.Consider the single linear equation 10×1 – 3×2 – 2×3 = 0 The nature of the solution to the above equation will be

The solution set is line through the origin

Q19.Square matrix A of order N over R has rank n, which of the following statements is not square?

A is Singular

Q20.If A and B are matrices and if AB is defined then the rank of AB equal to

<= min {rank A, rank B}

Q21.The vector Space R3 and its subspace H = {s,t,0} :s and t are real numbers) have the same dimensions


Q22.For a set of vectors to be considered a vector space what must be true about them?

the vectors are of varying lengths

Q23.Let A be an nxn invertible matrix. Which of the following statements is false.

the equation AX=0 has infinitely many solution

Q25. IF MSB bit is 1 the number is considered as

Negative Number

Q26.Let v be a vector space over the field F of dimension n. Consider the following statements
1.Every subset of V containing n elements is a basis of V.
2.No linearly independent subset of V contains more than n elements

Only 1

Q27. Let V be a five dimensional vector space and let S be a subset of V which spans V,Then S,

Must Be linearly dependent

Q28.For the set of equations x+2y+z+4w=2, 3x+6y+3x+12w=6. The following statement is true

Only the trivial solution x=y=z=0 exists

Q29.Every vector space has atleast two trivial subspaces.These are

(0) and V

Q30.The following set of vectors in R3 given by S ={v1,v2,v3} = {(1,2,3),(0,1,2),(-2,0,1)} is

linearly independent

Q31.Let p1(t) = 1, p2(t) = t, p3(t) = 4 – t. Then {p1(t), p2(t), p3(t)} is Linearly dependent in P = The set of polynomials, because :

a.  p1, p2, p3 span P.

b. None of the above

c. p3, = 4p1 – p2

d. p2(t) = 0 at t = 0

Ans: A

Q32.Let V be a vector space and W be non-empty  subset of V. If W is a vector space with respect to  operations  in V, then W is called

a. Basis of V

b. Subset of V

c. Inverse of V

d. Subspace of V

Ans: D

Q33.Consider the single linear equation, 10 x– 3 x2 – 2 x3  = 0 . The nature of solution to the above equation will be:

a. None of the above

b. The solution set is a plane through the origin

c. The solution set is a line through the origin

d. The equation has no solution

Ans: C

Q34.Do the vectors v1 = (−3, 7) and v2 = (5, 5) form a basis for R2 ?

a. Data not complete

b. Yes

c. No

d. None of the above

Ans: B

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