Vector Spaces, Subspaces and Bases Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Vector Spaces, Subspaces and Bases notes for GATE DA: 18 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

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.

Subspace Checklist: The Three Rules

A subset $W$ of a vector space $V$ is a subspace if and only if it passes these three checks:

Zero Vector

0 \in W

. The space must contain the origin.

Closed under Addition

u, v \in W

, then

u + v \in W

Closed under Scalar Multiplication

u \in W

and

c

is a scalar, then

c u \in W

Common Examples in $R^{n}$

Lines through origin
Yes, subspace

Planes through origin
Yes, subspace

Lines NOT through origin
Fails zero vector

First Quadrant ( $x \geq 0$ )
Fails scaling

Testing Linear Independence: The Rank Method

Given vectors $v_{1}, \dots, v_{k}$ in $R^{n}$ :

Step 1: Form matrix

A = [v_{1} ∣ v_{2} ∣ \dots ∣ v_{k}]

Step 2: Compute the Rank of

A

Decision Rule

If $Rank (A) = k$ : The vectors are Linearly Independent.

If $Rank (A) < k$ : The vectors are Linearly Dependent.

Special Case (Square Matrix)

If $k = n$ (square matrix), compute $det (A)$ :

$det (A) \neq = 0 ⟹$ Independent.
$det (A) = 0 ⟹$ Dependent.

More notes in this unit

chapter

Vector Spaces, Subspaces and Bases Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Vector Spaces, Subspaces and Bases notes for GATE DA: 18 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Free preview ends here

Login to view the complete notes

Creating an account is free. You get the rest of this chapter, step-by-step solutions, and a study plan built around the topics you are actually weak at.

Why MastersUp

Personalised first. High quality throughout.

Most platforms hand everyone the same content. Here the content moves with your performance, topic by topic.

Built around you, not around a syllabus PDF

Every answer you give moves your topic-level intelligence rate. The next question, the next revision card and tomorrow's plan all change with it.

Revision that hits your weak spots

We only revise topics you have actually attempted and are still below the safe bar on — never the same chapter on repeat.

Questions calibrated to the real exam

Each question carries a measured toughness. You are served a rung above your current level, so practice keeps stretching you.

Notes written for recall, not for volume

Full lesson cards for first study, curated short-note cards for the last mile — with derivations, traps and exam patterns marked.

One place for everything

Notes, chapter practice, previous-year questions, test series and full-length papers — all feeding one picture of your preparation.

Honest progress

No vanity streaks. Progress here means chapters mastered and accuracy that held up on harder questions.

Unlock the whole course

Full notes and short notes, the complete question bank with worked solutions, mock tests, full-length papers, and an adaptive plan that rebuilds itself as you improve.

View all plans Start free

Vector Spaces, Subspaces and Bases Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Chapter Roadmap: Vector Spaces, Subspaces and Bases

Chapter Journey: Vector Spaces & Bases

Hero Concept: The Basis as a Minimal Coordinate System

What is a Basis?

Why "Orthonormal" is the Gold Standard

Subspace Checklist: The Three Rules

Subspace Checklist: The Three Rules

Common Examples in Rn

Testing Linear Independence: The Rank Method

Testing Linear Independence: The Rank Method

Decision Rule

Special Case (Square Matrix)

More notes in this unit

Login to view the complete notes

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

Common Examples in $R^{n}$