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

Orthogonality, Projections and Linear Systems short notes for GATE DA: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Summary: Projection Matrix Master Sheet

Core Properties

P^{2} = P

Idempotent

P^{T} = P

Symmetric

Eigenvalues $λ \in {0, 1}$
$Trace (P) = Rank (P) = dim (C (P))$

Subspaces

$C (P) = U$ (Target subspace)
$N (P) = U^{⊥}$ (Orthogonal complement)
$R^{n} = C (P) \oplus N (P)$

Formulas

Line (vector $a$ ):

P = \frac{a a ^{T}}{a ^{T} a}

Subspace (matrix $A$ ):

P = A (A^{T} A)^{- 1} A^{T}

Complement: Projection onto

U^{⊥}

P^{⊥} = I - P

Summary: Consistency Master Sheet

Quick Summary

Consistency Master Sheet

Consistency Checklist

Consistent: $Rank (A) = Rank ([A ∣ b])$
Inconsistent: $Rank (A) < Rank ([A ∣ b])$

Counting Solutions (If Consistent)

Unique: $Rank (A) = n$
Infinite: $Rank (A) < n$ (Free variables = $n - Rank (A)$ )

Structure of Infinite Solutions

x = x_{p} + x_{h}

(Particular solution + Null space basis)

Golden Rules

$det (A) = 0$ does not mean no solution; it means not unique. Check $[A ∣ b]$ .
Free variables depend on $n$ (columns), not $m$ (rows).

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).

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

Orthogonality, Projections and Linear Systems short notes for GATE DA: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practi

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 ?

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