Orthogonality, Projections and Linear Systems Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Orthogonality, Projections and Linear Systems notes for GATE DA: 26 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Orthogonality, Projections and Linear Systems

Chapter Journey

Step 1: Projection Matrices, Null Space and Idempotence (Current)

Geometric intuition of orthogonal projections. Algebraic properties: Idempotence and Symmetry. Column space, Null space, and Eigenvalues.

Step 2: Consistency and Solution Sets of Linear Systems

Conditions for a linear system to have a solution. Geometric interpretation of consistency. Structuring the complete solution set.

Goal: Master the transition from projecting vectors to solving and analyzing linear systems.

The Geometry of Projection Matrices

The Core Idea

A projection matrix $P$ maps any vector $b \in R^{n}$ to a vector $p$ in a target subspace $U$ , such that the error vector $e = b - p$ is orthogonal to $U$ .

Geometric Meaning

$p$ is the "shadow" or closest point in $U$ .
$e$ is the "error" or perpendicular drop.
$P$ isolates the component of $b$ that lives inside $U$ .

The Two Golden Properties: Idempotence and Symmetry

1. Idempotence

P^{2} = P

Intuition: Projecting an already projected vector leaves it unchanged.
Algebra: $P (P b) = P p = p = P b$ .

2. Symmetry

P^{T} = P

Intuition: Guarantees the projection is orthogonal (perpendicular error).
Algebra: The error $b - P b$ is orthogonal to the column space of $P$ .

Definition Check: A matrix

P

is an orthogonal projection matrix if and only if

P^{2} = P

and

P^{T} = P

Column Space and Null Space of P

Space Decomposition

Let $P$ be the projection matrix onto subspace $U$ .

1. Column Space $C (P)$

$C (P) = U$
$P v = v$ for $v \in C (P)$

2. Null Space $N (P)$

$N (P) = U^{⊥}$
$P v = 0$ for $v \in N (P)$

Fundamental Theorem:

R^{n} = C (P) \oplus N (P)

Every vector

b = p + e

, where

p \in C (P)

and

e \in N (P)

Orthogonality, Projections and Linear Systems: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Linear Algebra MCQ

A linear system $A x = b$ is consistent if and only if:

A.
$Rank (A) = Rank ([A ∣ b])$
B.
$Rank (A) < Rank ([A ∣ b])$
C.
$det (A) \neq = 0$
D.
$b = 0$

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct-recall question about the fundamental condition for consistency.

Step 1: Consistency means there exists at least one solution $x$ .

Step 2: Geometrically, this means $b$ lies in the column space of $A$ .

Step 3: Algebraically, appending $b$ to $A$ to form the augmented matrix $[A ∣ b]$ adds a new column.

Step 4: If $b$ is already in the column space of $A$ , it does not increase the dimension of the column space. Thus, the rank remains unchanged.

Step 5: If $b$ is NOT in the column space, it adds a new independent direction, increasing the rank by 1.

Step 6: Therefore, consistency is equivalent to $Rank (A) = Rank ([A ∣ b])$ .

Answer: A

Question 2 · Linear Algebra MCQ

Which of the following pairs of algebraic properties uniquely defines an orthogonal projection matrix $P$ ?

A.
$P^{2} = I$ and $P^{T} = P$
B.
$P^{2} = P$ and $P^{T} = - P$
C.
$P^{2} = P$ and $P^{T} = P$
D.
$P^{2} = I$ and $P^{T} = - P$

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct recall question about the defining algebraic properties of orthogonal projection matrices.

Step 1: Recall the geometric meaning. An orthogonal projection drops a perpendicular to a target subspace.

Step 2: Algebraically, projecting a vector twice is the same as projecting it once, because the first projection already lands in the subspace. This gives idempotence: $P^{2} = P$ .

Step 3: The projection is "orthogonal", meaning the error vector is perpendicular to the subspace. This geometric requirement translates to the matrix being symmetric: $P^{T} = P$ .

Answer: $P^{2} = P$ and $P^{T} = P$ (Option C).

More notes in this unit

chapter

Orthogonality, Projections and Linear Systems Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Orthogonality, Projections and Linear Systems notes for GATE DA: 26 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice qu

A question from this chapter

Question 1

A linear system A x = b is consistent if and only if:

Question 2

Which of the following pairs of algebraic properties uniquely defines an orthogonal projection matrix P ?

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