Practice Problems on Unit-1
Each question may have one or more correct options
1. Which of the following is/are correct,
A. Every vector space contains a zero vector.
B. In any vector space, ax = by implies that a = b.
C. In any vector space, ax = ay implies that x = y.
D. A vector in Rn can be regarded as a matrix in Mn×1 (R).
2. Which of the following is/are correct,
A. V = {(x1 , x2 ) : x1 , x2 ∈ R} with addition defined coordinatewise and multiplication defined as
r(x1 , x2 ) = (x1 , 0) is a vector space.
B. V = {(x1 , x2 ) : x1 , x2 ∈ R} with addition defined as (x1 , x2 ) + (y1 , y2 ) = (x1 + 2y2 , x2 + 3y2 )
and multiplication defined as r(x1 , x2 ) = (rx1 , rx2 ) is a vector space.
C. V = {(x1 , x2 ) : x1 , x2 ∈ R} with addition defined coordinatewise and multiplication defined as
(
(0, 0) if r = 0
r(x1 , x2 ) =
(rx1 , xr2 ) if r 6= 0
is a vector space.
3. Which of the following is/are subspaces of R3 ,
A. W1 = {(x, y, z) ∈ R : x = 3y and z = −y}.
B. W2 = {(x, y, z) ∈ R : x = z + 2}.
C. W3 = {(x, y, z) ∈ R : 2x − 7y + z = 0}.
D. W1 = {(x, y, z) ∈ R : 5x2 − 3y 2 + 6z 2 = 0}. [3 Marks]
4. Which of the following is/are correct,
A. The span of φ is φ.
B. The zero vector is a linear combination of any nonempty set of vectors.
C. −x3 + 2x2 + 3x + 3 ∈ span{x3 + x2 + x + 1, x2 + x + 1, x + 1}.
D. (2, 1, 1) ∈ span{(1, 0, 2), (−1, 1, 1)}.
5. Which of the following is/are true,
1 −3 2 −3 7 4 −2 3 11
A. In M2×3 (R), the set , , is linearly dependent.
−4 0 5 6 −2 −7 −1 −3 2
B. In R4 the set {(1, 0, 0, −1), (0, 1, 0, −1), (0, 0, 1, −1), (0, 0, 0, 1)} is linearly independent.
C. Any set containing the zero vector is linearly dependent.
D. Subsets of linearly dependent sets are linearly dependent.
6. Determine which of the following is/are correct
A. The vector space {0} has dimension zero.
B. {x2 + 3x − 2, 2x2 + 5x − 3, −x2 − 4x + 4} is a basis for P2 (R).
C. A basis for zero vector space is the empty set.
D. W = {(a1 , a2 , a3 , a4 , a5 ) ∈ R5 : a1 + a3 + a5 = 0, a2 = a4 } is a subspace of R5 having
{(−1, 0, 1, 0, 0), (−1, 0, 0, 0, 1), (0, 1, 0, 1, 0)} as a basis and dim(W ) = 3.
7. Which of the following is/are correct.
A. A linear transformation T : V → W is one-one iff N (T ) = {0}.
B. T : P2 (R) → P3 (R) defined by
Z x
0
T (p(x)) = 2p (x) + 3p(t)dt
0
is one-one.
C. For the linear transformation T : P2 (R) → P3 (R) defined by
Z x
T (p(x)) = 2p0 (x) + 3p(t)dt
0
we have rank = dim(R(T )) = 3.
D. For the linear transformation T : M2×3 (R) → M2×2 (R) defined by
a11 a12 a13 2a11 − a12 a13 + 2a12
T =
a21 a22 a23 0 0
rank=2 and nullity=4.
8. Which of the following is/are correct.
A. The triangle inequality only holds in finite-dimensional inner product spaces.
B. An inner product is linear in both components.
C. If x, y, and z are vectors in an inner product space such that hx, yi = hx, zi then y = z.
D. If hx, yi = 0 for all x in an inner product space, then y = 0.
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