chapter

Vector Spaces, Rank, Nullity and Orthogonality Notes for GATE CS

Vector Spaces, Rank, Nullity and Orthogonality notes for GATE CS: 30 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice q

vector spaces rank nullity and orthogonality notes

Chapter Roadmap: Vector Spaces, Rank, Nullity and Orthogonality

Chapter Roadmap Vector Spaces, Rank, Nullity and Orthogonality Master the geometry of linear equations, from solution spaces to orthogonal bases. Topic 1 \cdot High Weightage Homogeneous Systems and Nontrivial Null Spaces Understand when systems have infinite solutions and define the null space. Topic 2 Rank-Nullity Computations Compute dimensions of row, column, and null spaces using the fundamental theorem. Topic 3 Orthogonality and Maximum Orthogonal Sets Find orthogonal bases and understand the limits of orthogonal sets in n-dimensional space.

Homogeneous Systems and Nontrivial Null Spaces

Vector Spaces \cdot Topic 1 Homogeneous Systems & Nontrivial Null Spaces Move beyond just solving equations to understanding the geometry of solutions. Determine exactly when a system has solutions other than zero. Identify the null space and use its subspace properties. Crack tricky exam questions using standard basis vectors.

The Homogeneous System Ax = 0

The Homogeneous System A homogeneous system of linear equations is written in matrix form as: A x = 0 where A is an m \times n matrix and x is a column vector of n variables. The Golden Rule: A homogeneous system is always consistent . It always has at least one solution: the trivial solution x = 0 .

27 more cards in this chapter

Try a question

Answer it here to see how it works. Nothing is recorded until you sign in.

Question 1

Level 1: Warm-up

What is the maximum number of non-zero, mutually orthogonal vectors that can exist in R^{4} ?

Question 2

Level 1: Warm-up

What is the maximum number of non-zero, mutually orthogonal vectors that can exist in R^{7} ?

Question 3

Level 1: Warm-up

Let A be an m \times n matrix. According to the Rank-Nullity Theorem, which of the following equations correctly relates the rank and nullity of A ?

Question 4

Level 1: Warm-up

Two non-zero vectors u and v in R^{n} are orthogonal if and only if:

Question 5

Level 1: Warm-up

Let S be a set of non-zero vectors in R^{n} that are mutually orthogonal. Which of the following properties is ALWAYS guaranteed for S ?

Question 6

Level 1: Warm-up

Let A be a 4 \times 6 matrix. If the rank of A is 3, what is the nullity of A ?

Question 7

Level 1: Warm-up

For a square n \times n matrix A, which of the following conditions guarantees that the homogeneous system A x = 0 has a non-trivial solution?

Question 8

Level 1: Warm-up

Consider the matrix A = [13 2 k] . For what value of k does the homogeneous system A x = 0 have a non-trivial solution?

Question 9

Level 1: Warm-up

Let A be a 3 \times 5 matrix. Which of the following CANNOT be the rank of A ?

Question 10

Level 1: Warm-up

Which of the following pairs of vectors in R^{3} is orthogonal?

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, Rank, Nullity and Orthogonality Notes for GATE CS

Vector Spaces, Rank, Nullity and Orthogonality notes for GATE CS: 30 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Vector Spaces, Rank, Nullity and Orthogonality

Chapter Roadmap

Vector Spaces, Rank, Nullity and Orthogonality

Master the geometry of linear equations, from solution spaces to orthogonal bases.

Topic 1 · High Weightage

Homogeneous Systems and Nontrivial Null Spaces

Understand when systems have infinite solutions and define the null space.

Topic 2

Rank-Nullity Computations

Compute dimensions of row, column, and null spaces using the fundamental theorem.

Topic 3

Orthogonality and Maximum Orthogonal Sets

Find orthogonal bases and understand the limits of orthogonal sets in n-dimensional space.

Homogeneous Systems and Nontrivial Null Spaces

Vector Spaces · Topic 1

Homogeneous Systems & Nontrivial Null Spaces

Move beyond just solving equations to understanding the geometry of solutions.

Determine exactly when a system has solutions other than zero.
Identify the null space and use its subspace properties.
Crack tricky exam questions using standard basis vectors.

The Homogeneous System Ax = 0

The Homogeneous System

A homogeneous system of linear equations is written in matrix form as:

A x = 0

where $A$ is an $m \times n$ matrix and $x$ is a column vector of $n$ variables.

The Golden Rule:

A homogeneous system is always consistent. It always has at least one solution: the trivial solution $x = 0$ .

When Does a System Have More Than Just Zero?

Beyond the Trivial Solution

While the trivial solution always exists, exams focus on nontrivial solutions — any solution where $x \neq = 0$ .

Geometric Meaning:

Nontrivial solutions exist if and only if the columns of $A$ are linearly dependent. This means there is some non-zero combination of the columns that adds up to the zero vector.

In simple terms: The system has "free variables" that can take on any value, generating infinitely many solutions.

Vector Spaces, Rank, Nullity and Orthogonality: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Engineering Mathematics MCQ

What is the maximum number of non-zero, mutually orthogonal vectors that can exist in $R^{4}$ ?

A.
2
B.
3
C.
4
D.
8

Correct Answer:

Step-by-Step Solution

Key idea: This is a "maximum orthogonal set" question, recognisable because it asks for the maximum cardinality of a set of pairwise orthogonal non-zero vectors in a specific dimension.

Step 1: Recall the key theorem. Any set of non-zero, mutually orthogonal vectors in $R^{n}$ is linearly independent.

Step 2: Apply the dimension bound. In $R^{n}$ , you cannot have more than $n$ linearly independent vectors. Therefore, the maximum number of non-zero mutually orthogonal vectors is exactly $n$ .

Step 3: Substitute the given dimension. For $R^{4}$ , the maximum number is $4$ .

Answer: 4

Question 2 · Engineering Mathematics MCQ

What is the maximum number of non-zero, mutually orthogonal vectors that can exist in $R^{7}$ ?

A.
5
B.
6
C.
7
D.
8

Correct Answer:

Step-by-Step Solution

Key idea: This is a "maximum orthogonal set size" question, recognisable because it asks for the maximum cardinality of a set of pairwise orthogonal non-zero vectors in a specific dimensional space.

Step 1: Recall the link between orthogonality and independence. Non-zero mutually orthogonal vectors are linearly independent.

Step 2: Recall the dimension bound. In $R^{n}$ , you cannot have more than $n$ linearly independent vectors.

Step 3: Apply to the given space. For $R^{7}$ , the maximum number of linearly independent vectors is 7. Therefore, the maximum number of non-zero mutually orthogonal vectors is exactly 7.

Answer: 7

Question 3 · Engineering Mathematics MCQ

Let $A$ be an $m \times n$ matrix. According to the Rank-Nullity Theorem, which of the following equations correctly relates the rank and nullity of $A$ ?

A.
$e x t r ank (A) + e x t n u l l i t y (A) = m$
B.
$e x t r ank (A) + e x t n u l l i t y (A) = n$
C.
$e x t r ank (A) - e x t n u l l i t y (A) = n$
D.
$e x t r ank (A) im ese x t n u l l i t y (A) = mim es n$

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct recall question about the statement of the Rank-Nullity Theorem.

Step 1: Recall the Rank-Nullity Theorem. It states that for an $m \times n$ matrix $A$ , the sum of the rank and the nullity equals the number of columns.

Step 2: Identify the number of columns. For an $m \times n$ matrix, the number of columns is $n$ .

Step 3: Write the equation: $rank (A) + nullity (A) = n$ .

Answer: B

Question 4 · Engineering Mathematics MCQ

Two non-zero vectors $u$ and $v$ in $R^{n}$ are orthogonal if and only if:

A.
u · v = 1
B.
u · v = 0
C.
||u|| = ||v||
D.
u × v = 0

Correct Answer:

Step-by-Step Solution

Key idea: This is a "definition of orthogonality" question, recognisable because it asks for the algebraic condition that defines when two vectors are perpendicular.

Step 1: Recall the geometric meaning of orthogonality. Two vectors are orthogonal if the angle between them is $9 0^{\circ}$ .

Step 2: Recall the algebraic definition. The dot product $u \cdot v = ∥ u ∥∥ v ∥ cos (θ)$ . If $θ = 9 0^{\circ}$ , then $cos (9 0^{\circ}) = 0$ , so $u \cdot v = 0$ .

Step 3: Evaluate the options. The condition $u \cdot v = 0$ is the exact definition of orthogonality.

Answer: u · v = 0

Question 5 · Engineering Mathematics MCQ

Let $S$ be a set of non-zero vectors in $R^{n}$ that are mutually orthogonal. Which of the following properties is ALWAYS guaranteed for $S$ ?

A.
$S$ spans the entire space $R^{n}$ .
B.
Every vector in $S$ is a unit vector.
C.
$S$ is a linearly independent set.
D.
$S$ contains exactly $n$ vectors.

Correct Answer:

Step-by-Step Solution

Key idea: This is an "orthogonal sets and linear independence" question, recognisable because it describes a set of mutually orthogonal non-zero vectors and asks for a guaranteed structural property.

Step 1: Recall the fundamental theorem. Any set of non-zero, pairwise orthogonal vectors is automatically linearly independent.

Step 2: Evaluate the other options to see why they are not guaranteed.

Spanning $R^{n}$ : Not guaranteed. $S$ could just be a single vector, or a subset of a basis.
Unit vectors: Not guaranteed. Orthogonal vectors can have any non-zero magnitude.
Exactly $n$ vectors: Not guaranteed. The set could have fewer than $n$ vectors (e.g., just 2 orthogonal vectors in $R^{5}$ ).

Step 3: Conclude. Linear independence is the only property strictly guaranteed by the given conditions.

Answer: $S$ is a linearly independent set.

Question 6 · Engineering Mathematics MCQ

Let $A$ be a $4 \times 6$ matrix. If the rank of $A$ is $3$ , what is the nullity of $A$ ?

A.
1
B.
2
C.
3
D.
4

Correct Answer:

Step-by-Step Solution

Key idea: This is a "rank-nullity computation" question, recognisable because it gives the dimensions of the matrix and the rank, and asks for the nullity.

Step 1: Recall the Rank-Nullity Theorem. For any $m \times n$ matrix $A$ , the sum of the rank and the nullity equals the number of columns $n$ : $rank (A) + nullity (A) = n$ .

Step 2: Identify the number of columns. The matrix $A$ is $4 \times 6$ , so it has $n = 6$ columns.

Step 3: Solve for the nullity. We are given $rank (A) = 3$ . Substituting into the theorem: $3 + nullity (A) = 6 ⟹ nullity (A) = 3$ .

Answer: 3

Question 7 · Engineering Mathematics MCQ

For a square $n \times n$ matrix $A$ , which of the following conditions guarantees that the homogeneous system $A x = 0$ has a non-trivial solution?

A.
$\det(A)

eq 0$
B.
$det (A) = 0$
C.
$e x t r ank (A) = n$
D.
The rows of $A$ are linearly independent.

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct recall question about the determinant condition for non-trivial solutions in square homogeneous systems.

Step 1: Recall the property of homogeneous systems. The system $A x = 0$ always has the trivial solution $x = 0$ .

Step 2: For a square matrix $A$ , a non-trivial solution exists if and only if the columns of $A$ are linearly dependent.

Step 3: The columns of $A$ are linearly dependent if and only if the determinant of $A$ is zero.

Therefore, the condition that guarantees a non-trivial solution is $det (A) = 0$ .

Answer: B

Question 8 · Engineering Mathematics MCQ

Consider the matrix $A = [13 2 k]$ . For what value of $k$ does the homogeneous system $A x = 0$ have a non-trivial solution?

A.
5
B.
6
C.
7
D.
8

Correct Answer:

Step-by-Step Solution

Key idea: This is a "determinant condition for non-trivial solutions" question, recognisable because it asks for a parameter that makes a square homogeneous system have non-trivial solutions.

Step 1: Recall the condition for non-trivial solutions. For a square matrix $A$ , the system $A x = 0$ has a non-trivial solution if and only if $det (A) = 0$ .

Step 2: Compute the determinant of $A$ . $det (A) = (1) (k) - (2) (3) = k - 6$ .

Step 3: Set the determinant to zero and solve for $k$ . $k - 6 = 0 ⟹ k = 6$ .

Answer: 6

Question 9 · Engineering Mathematics MCQ

Let $A$ be a $3 \times 5$ matrix. Which of the following CANNOT be the rank of $A$ ?

A.
1
B.
2
C.
3
D.
4

Correct Answer:

Step-by-Step Solution

Key idea: This is a "rank boundary" question, recognisable because it asks for an impossible value for the rank of a matrix with given dimensions.

Step 1: Recall the bounds on the rank of a matrix. For any $m \times n$ matrix $A$ , the rank is bounded by the minimum of its dimensions: $0 \leq rank (A) \leq min (m, n)$ .

Step 2: Apply the bounds to the given matrix. The matrix $A$ is $3 \times 5$ , so $m = 3$ and $n = 5$ . The maximum possible rank is $min (3, 5) = 3$ .

Step 3: Evaluate the options. The rank can be 0, 1, 2, or 3. It cannot be 4, because that would exceed the number of rows.

Answer: 4

Question 10 · Engineering Mathematics MCQ

Which of the following pairs of vectors in $R^{3}$ is orthogonal?

A.
$[1, 2, 3]$ and $[2, 4, 6]$
B.
$[1, 0, - 1]$ and $[1, 2, 1]$
C.
$[1, 1, 1]$ and $[- 1, - 1, - 1]$
D.
$[0, 1, 0]$ and $[0, 2, 0]$

Correct Answer:

Step-by-Step Solution

Key idea: This is an "orthogonality check via dot product" question, recognisable because it asks to identify a perpendicular pair of vectors from a list of options.

Step 1: Recall the condition for orthogonality. Two vectors $u$ and $v$ are orthogonal if and only if their dot product $u \cdot v = 0$ .

Step 2: Compute the dot product for each option.

Option A: $(1) (2) + (2) (4) + (3) (6) = 2 + 8 + 18 = 28 \neq = 0$ .
Option B: $(1) (1) + (0) (2) + (- 1) (1) = 1 + 0 - 1 = 0$ .
Option C: $(1) (- 1) + (1) (- 1) + (1) (- 1) = - 1 - 1 - 1 = - 3 \neq = 0$ .
Option D: $(0) (0) + (1) (2) + (0) (0) = 0 + 2 + 0 = 2 \neq = 0$ .

Step 3: Conclude. Only the pair in Option B has a dot product of zero.

Answer: $[1, 0, - 1]$ and $[1, 2, 1]$

vector spaces rank nullity and orthogonality notes

Chapter Roadmap: Vector Spaces, Rank, Nullity and Orthogonality

Vector Spaces, Rank, Nullity and Orthogonality

Homogeneous Systems and Nontrivial Null Spaces

Homogeneous Systems & Nontrivial Null Spaces

The Homogeneous System Ax = 0

The Homogeneous System

Try a question

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

Vector Spaces, Rank, Nullity and Orthogonality Notes for GATE CS

Chapter Roadmap: Vector Spaces, Rank, Nullity and Orthogonality

Vector Spaces, Rank, Nullity and Orthogonality

Homogeneous Systems and Nontrivial Null Spaces

Homogeneous Systems & Nontrivial Null Spaces

The Homogeneous System Ax = 0

The Homogeneous System

When Does a System Have More Than Just Zero?

Beyond the Trivial Solution

Vector Spaces, Rank, Nullity and Orthogonality: Solved Questions with Step-by-Step Explanations (10 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

More notes in this unit