Vector Spaces, Subspaces and Bases Previous Year Questions (PYQs) for GATE DA: 3+ Solved Questions with Step-by-Step Solutions

Solve 3+ Vector Spaces, Subspaces and Bases previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Vector Spaces, Subspaces and Bases

Chapter Journey: Vector Spaces & Bases

Vector Spaces & Subspaces

Defining the playground. What makes a set a space?

Linear Independence

Removing redundancy. Do we really need all these vectors?

Bases & Orthonormal Bases

The minimal building blocks. Constructing perfect coordinate systems.

Geometry of Subspaces

Visualizing norms, balls, and intersections in higher dimensions.

Why this matters for GATE DA: Understanding bases allows you to change perspectives, simplify matrices, and compress data. This topic is the foundation for almost all advanced linear algebra applications in data science.

Hero Concept: The Basis as a Minimal Coordinate System

What is a Basis?

A basis for a vector space $V$ is a sequence of vectors $B = {v_{1}, v_{2}, \dots, v_{n}}$ that satisfies two critical conditions:

Linear Independence: No vector in the set can be written as a combination of the others. There is no "waste" or redundancy.
Spanning: Every vector in $V$ can be written as a linear combination of vectors in $B$ .

\forall v \in V, \exists c_{1}, \dots, c_{n} such that v = c_{1} v_{1} + \dots + c_{n} v_{n}

Why "Orthonormal" is the Gold Standard

An orthonormal basis is a basis where:

Orthogonal: All vectors are perpendicular to each other ( $v_{i} \cdot v_{j} = 0$ for $i \neq = j$ ).
Normal: Each vector has length 1 ( $∥ v_{i} ∥ = 1$ ).

The Superpower

If ${u_{1}, \dots, u_{n}}$ is an orthonormal basis, finding the coefficients $c_{i}$ for any vector $v$ becomes trivial:

c_{i} = v \cdot u_{i}

You do not need to solve a system of linear equations. You just project $v$ onto each basis vector. This is the core idea behind Fourier series, PCA, and many signal processing techniques.

Vector Spaces, Subspaces and Bases: Solved Questions with Step-by-Step Explanations (3 Problems)

Question 1 · Linear Algebra MSQ

Which of the following statements is/are correct?

A.
$R^{n}$ has a unique set of orthonormal basis vectors
B.
$R^{n}$ does not have a unique set of orthonormal basis vectors
C.
Linearly independent vectors in $R^{n}$ are orthonormal
D.
Orthonormal vectors $R^{n}$ are linearly independent

Question 2 · Linear Algebra MCQ

Consider a set

S_{1} = {x = (x_{1}, x_{2}, x_{3})^{T} \in R^{3} ∣ x^{T} x \leq 16}

. Let

S_{2}

be another set which is a subspace of

R^{3}

with dimension two.
Which of the following gives the area of

S_{1} \cap S_{2}

A.
$16 π$
B.
$4 π$
C.
$4 π^{2}$
D.
$16 π^{2}$

Question 3 · Linear Algebra MSQ

Select all choices that are subspaces of

R^{3}

.
Note:

R

denotes the set of real numbers.

A.
${x = [x_{1} \x_{2} \x_{3}] \in R^{3} : x = α 110 + β 100, α, β \in R}$
B.
${x = [x_{1} \x_{2} \x_{3}] \in R^{3} : x = α^{2} 120 + β^{2} 101, α, β \in R}$
C.
${x = [x_{1} \x_{2} \x_{3}] \in R^{3} : 5 x_{1} + 2 x_{3} = 0, 4 x_{1} - 2 x_{2} + 3 x_{3} = 0}$
D.
${x = [x_{1} \x_{2} \x_{3}] \in R^{3} : 5 x_{1} + 2 x_{3} + 4 = 0}$

A question from this chapter

Question 1

Which of the following statements is/are correct?

Question 2

Consider a set S_{1} = {x = (x_{1}, x_{2}, x_{3})^{T} \in R^{3} ∣ x^{T} x \leq 16} . Let S_{2} be another set which is a subspace of R^{3} with dimension two. Which of the following gives the area of S_{1} \cap S_{2} ?

Question 3

Select all choices that are subspaces of R^{3} . Note: R denotes the set of real numbers.

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