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Vector Spaces PYQs for GATE DA

GATE DA Vector Spaces: 5 chapters, 5 previous year questions (25% of Linear Algebra), 266 practice questions and one solved question from each chapter.

A question from this chapter

Question 1

2026 PYQ

Level 3: Exam Standard

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 2

2024 PYQ

Level 3: Exam Standard

Which of the following statements is/are TRUE? Note: R denotes the set of real numbers.

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Vector Spaces PYQs for GATE DA

GATE DA Vector Spaces: 5 chapters, 5 previous year questions (25% of Linear Algebra), 266 practice questions and one solved question from each chapter.

About Vector Spaces Previous Year Questions (PYQs)

5 previous year questions from Vector Spaces in GATE DA, grouped by chapter with the exam year, answer key and step-by-step solution for each.

Vector Spaces Weightage in GATE DA

Vector Spaces accounts for 5 of 20 Linear Algebra previous year questions in our bank (25%), about 1.7 per paper across 3 papers.

Vector Spaces Chapter Matrix

Chapter	Topics	PYQs	Share of unit PYQs	Practice questions
Chapter 1 — Vector Spaces		0	0%	0
Vector Spaces, Subspaces and Bases	Subspaces, Linear Independence and Orthonormal Bases, Geometry of Subspaces and Norm Balls	3	60%	154
Chapter 2 — Vector Spaces		0	0%	0
Orthogonality, Projections and Linear Systems	Projection Matrices, Null Space and Idempotence, Consistency and Solution Sets of Linear Systems	2	40%	112
Chapter 3 — Vector Spaces		0	0%	0

One Solved Question from Each Vector Spaces Chapter

Question 1 · Vector Spaces, Subspaces and Bases · 2026 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.
$16 π^{2}$

Correct Answer:

Step-by-Step Solution

Insight: The intersection of a solid n-dimensional ball with a k-dimensional subspace passing through the origin is a solid k-dimensional ball of the exact same radius.

Exam route: $x^{T} x \leq 16$ means $∥ x ∥_{2} \leq 4$ , so it's a solid 3D ball of radius $R = 4$ . $S_{2}$ is a 2D subspace (a plane through the origin). The intersection is a 2D solid disk of radius 4. Area = $π R^{2} = 16 π$ .

Learning route:

Identify $S_{1}$ : The condition $x^{T} x \leq 16$ is equivalent to $∥ x ∥_{2}^{2} \leq 16$ , which means $∥ x ∥_{2} \leq 4$ . This defines a solid 3-dimensional $L_{2}$ ball centered at the origin with radius $R = 4$ .
Identify $S_{2}$ : A subspace of $R^{3}$ with dimension 2 is a flat plane that passes exactly through the origin.
Find the intersection $S_{1} \cap S_{2}$ : Slicing a solid 3D sphere with a plane that passes through its exact center (the origin) yields a solid 2D disk (a "great disk") with the same radius $R = 4$ .
Calculate the area: The area of a 2D disk of radius $R$ is $π R^{2}$ . Substituting $R = 4$ , we get Area = $π (4)^{2} = 16 π$ .

Question 2 · Orthogonality, Projections and Linear Systems · 2024 MSQ

Which of the following statements is/are TRUE?
Note:

R

denotes the set of real numbers.

A. There exist $M \in R^{3 \times 3}$ , $p \in R^{3}$ , and $q \in R^{3}$ such that $M x = p$ has a unique
solution and $M x = q$ has infinite solutions.
B. There exist $M \in R^{3 \times 3}$ , $p \in R^{3}$ , and $q \in R^{3}$ such that $M x = p$ has no solutions
and $M x = q$ has infinite solutions.
C. There exist $M \in R^{2 \times 3}$ , $p \in R^{2}$ , and $q \in R^{2}$ such that $M x = p$ has a unique
solution and $M x = q$ has infinite solutions.
D. There exist $M \in R^{3 \times 2}$ , $p \in R^{3}$ , and $q \in R^{3}$ such that $M x = p$ has a unique
solution and $M x = q$ has no solutions.

Correct Answer:

["B","D"]

Step-by-Step Solution

Key idea: This question tests the consistency conditions of linear systems $A x = b$ based on the shape of matrix $A$ (square, wide, or tall) and its rank.

Step 1: Analyze Option A ( $M \in R^{3 \times 3}$ ).

For a square matrix $M$ , if $M x = p$ has a unique solution, then $det (M) \neq = 0$ ( $M$ is invertible).

If $M$ is invertible, $M x = q$ must also have a unique solution ( $x = M^{- 1} q$ ) for ANY $q$ .

It is impossible for an invertible matrix to have infinite solutions for any RHS.

Thus, Option A is FALSE.

Step 2: Analyze Option B ( $M \in R^{3 \times 3}$ ).

Can $M x = p$ have no solution and $M x = q$ have infinite solutions?

Yes. Let $M$ be singular (Rank < 3).

Example: $M = 100010000$ .

Let $p = 001$ . System is inconsistent (no solution) because $0 \neq = 1$ .
Let $q = 000$ . System is $x = 0, y = 0, 0 = 0$ . $z$ is free. Infinite solutions.

Thus, Option B is TRUE.

Step 3: Analyze Option C ( $M \in R^{2 \times 3}$ ).

Can $M x = p$ have a unique solution?

$M$ has 3 columns (variables) and 2 rows (equations).

Max Rank is 2.

Number of free variables = $3 - Rank (M) \geq 3 - 2 = 1$ .

If a solution exists, there is at least 1 free variable, implying infinite solutions.

A unique solution is impossible for a wide matrix ( $n > m$ ).

Thus, Option C is FALSE.

Step 4: Analyze Option D ( $M \in R^{3 \times 2}$ ).

Can $M x = p$ have a unique solution?

$M$ has 2 columns. Max Rank is 2.

If Rank( $M$ )=2, the columns are linearly independent.

If $p$ is in the column space, the solution is unique (0 free variables).

Can $M x = q$ have no solution?

Yes, if $q$ is not in the column space (which is a 2D plane in $R^{3}$ ).

So, Unique Solution AND No Solution are both possible for different RHS vectors.

Thus, Option D is TRUE.

Answer: Options B and D are TRUE.

A question from this chapter

Login to view the complete previous-year questions and solutions

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Vector Spaces PYQs for GATE DA

About Vector Spaces Previous Year Questions (PYQs)

Vector Spaces Weightage in GATE DA

Vector Spaces Chapter Matrix

More from Linear Algebra

One Solved Question from Each Vector Spaces Chapter

Step-by-Step Solution

Step-by-Step Solution