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

Vector Spaces, Rank, Nullity and Orthogonality short notes for GATE CS: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved pract

vector spaces rank nullity and orthogonality short notes

Summary: The Homogeneous System Checklist

Summary Checklist The Homogeneous Execution Plan 1 The Baseline A x = 0 always has the trivial solution x = 0 . 2 Square Matrices (m = n) Nontrivial solutions exist ⟺ det (A) = 0 ⟺ rank (A) < n . 3 Rectangular Matrices (m \neq = n) If variables exceed equations (n > m), nontrivial solutions are guaranteed. Otherwise, check if rank (A) < n . 4 The Null Space N (A) is a subspace of R^{n} . It is closed under addition and scalar multiplication. 5 The Basis Vector Shortcut If a standard basis vector e_{k} \in N (A), the entire k -th column of A is exactly zero.

Summary: The Rank-Nullity Checklist

Summary Checklist The Rank-Nullity Execution Plan 1 Identify dimensions Matrix is m \times n . Remember: rank + nullity = n . 2 If you have the matrix Row reduce \to count pivots (rank) and non-pivots (nullity). 3 If null space is described Count basis vectors \to that is nullity. 4 Apply the theorem rank = n - nullity (or vice versa). 5 Verify the bound Check: rank \leq min (m, n) .

Quick Revision: Orthogonality Limits

Quick Revision: Orthogonality Limits Summary Checklist 1 Definition: u ⊥ v ⟺ u \cdot v = 0 . 2 Independence: Any set of non-zero pairwise orthogonal vectors is Linearly Independent. 3 Maximum Limit: In R^{n}, max number of non-zero mutually orthogonal vectors is n . 4 Exam Pattern: Identify "sum of products" as the dot product \to Check dimensions \to Check "non-zero" constraint \to Answer is the dimension n .

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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?

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

Vector Spaces, Rank, Nullity and Orthogonality short notes for GATE CS: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Summary: The Homogeneous System Checklist

Summary Checklist

The Homogeneous Execution Plan

The Baseline

$A x = 0$ always has the trivial solution $x = 0$ .

Square Matrices ( $m = n$ )

Nontrivial solutions exist $⟺ det (A) = 0 ⟺ rank (A) < n$ .

Rectangular Matrices ( $m \neq = n$ )

If variables exceed equations ( $n > m$ ), nontrivial solutions are guaranteed. Otherwise, check if $rank (A) < n$ .

The Null Space

$N (A)$ is a subspace of $R^{n}$ . It is closed under addition and scalar multiplication.

The Basis Vector Shortcut

If a standard basis vector $e_{k} \in N (A)$ , the entire $k$ -th column of $A$ is exactly zero.

Summary: The Rank-Nullity Checklist

Summary Checklist

The Rank-Nullity Execution Plan

Identify dimensions

Matrix is $m \times n$ . Remember: rank + nullity = $n$ .

If you have the matrix

Row reduce → count pivots (rank) and non-pivots (nullity).

If null space is described

Count basis vectors → that is nullity.

Apply the theorem

rank = $n$ - nullity (or vice versa).

Verify the bound

Check: rank $\leq min (m, n)$ .

Quick Revision: Orthogonality Limits

Summary Checklist

1 Definition: $u ⊥ v ⟺ u \cdot v = 0$ .
2 Independence: Any set of non-zero pairwise orthogonal vectors is Linearly Independent.
3 Maximum Limit: In $R^{n}$ , max number of non-zero mutually orthogonal vectors is $n$ .
4 Exam Pattern: Identify "sum of products" as the dot product $\to$ Check dimensions $\to$ Check "non-zero" constraint $\to$ Answer is the dimension $n$ .

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 short notes

Summary: The Homogeneous System Checklist

The Homogeneous Execution Plan

Summary: The Rank-Nullity Checklist

The Rank-Nullity Execution Plan

Quick Revision: Orthogonality Limits

Quick Revision: Orthogonality Limits

Summary Checklist

Try a question

Login to view the complete short 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 Short Notes for GATE CS

Summary: The Homogeneous System Checklist

The Homogeneous Execution Plan

Summary: The Rank-Nullity Checklist

The Rank-Nullity Execution Plan

Quick Revision: Orthogonality Limits

Quick Revision: Orthogonality Limits

Summary Checklist

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

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