Vector Spaces, Subspaces and Bases Practice Questions for GATE DA

Solve 154+ Vector Spaces, Subspaces and Bases practice questions for GATE DA with answers and detailed solutions. Free sample questions below.

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.

Vector Spaces, Subspaces and Bases: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Linear Algebra MCQ

Assertion (A): In the Gram-Schmidt process, the first step to find $u_{1}$ is to normalize the initial vector $v_{1}$ .

Reason (R): Normalization ensures that $u_{1}$ is orthogonal to all other vectors in the basis.

A.
Both A and R are true, and R is the correct explanation of A.
B.
Both A and R are true, but R is not the correct explanation of A.
C.
A is true, but R is false.
D.
A is false, but R is true.

Correct Answer:

Step-by-Step Solution

Key idea: Distinguish between normalization (unit length) and orthogonalization (perpendicularity).

Step 1: Evaluate Assertion (A). The Gram-Schmidt process starts by taking $v_{1}$ and dividing by its norm to get $u_{1}$ . This is normalization. So, A is true.

Step 2: Evaluate Reason (R). Normalization only ensures the vector has a length of 1 ( $∥ u_{1} ∥ = 1$ ). It does not make the vector orthogonal to anything. Orthogonality is achieved in subsequent steps by subtracting projections. So, R is false.

Step 3: Match with options. A is true, R is false.

Answer: A is true, but R is false.

Question 2 · Linear Algebra MCQ

Assertion (A): In the Gram-Schmidt process, the vector $w_{2}$ is constructed by subtracting the projection of $v_{2}$ onto $u_{1}$ from $v_{2}$ .

Reason (R): This subtraction step ensures that the resulting vector $w_{2}$ has a unit length.

A.
Both A and R are true, and R is the correct explanation of A.
B.
Both A and R are true, but R is not the correct explanation of A.
C.
A is true, but R is false.
D.
A is false, but R is true.

Correct Answer:

Step-by-Step Solution

Key idea: Distinguish between orthogonalization (subtraction) and normalization (division by norm). Step 1: Evaluate Assertion (A). The formula for

w_{2}

is indeed

w_{2} = v_{2} - (v_{2} \cdot u_{1}) u_{1}

. This is correct. Step 2: Evaluate Reason (R). The subtraction step removes the component of

v_{2}

that is parallel to

u_{1}

, making

w_{2}

orthogonal to

u_{1}

. It does not ensure unit length. Normalization (

u_{2} = w_{2} /∥ w_{2} ∥

) is required for unit length. Step 3: Conclude that A is true, but R is false. Answer: A is true, but R is false.

Question 3 · Linear Algebra MCQ

Assertion (A): In the Gram-Schmidt process, the final step to find $u_{2}$ is to divide the orthogonalized vector $w_{2}$ by its norm $∥ w_{2} ∥$ .

Reason (R): This division step ensures that the resulting vector $u_{2}$ is orthogonal to $u_{1}$ .

A.
Both A and R are true, and R is the correct explanation of A.
B.
Both A and R are true, but R is not the correct explanation of A.
C.
A is true, but R is false.
D.
A is false, but R is true.

Correct Answer:

Step-by-Step Solution

Key idea: Distinguish between orthogonalization (subtraction) and normalization (division by norm). Step 1: Evaluate Assertion (A). The formula for

u_{2}

is indeed

u_{2} = w_{2} /∥ w_{2} ∥

. This is correct. Step 2: Evaluate Reason (R). The division step scales

w_{2}

to have a length of 1. It does not ensure orthogonality. Orthogonality was already achieved in the previous step when

w_{2}

was constructed by subtracting the projection of

v_{2}

onto

u_{1}

. Step 3: Conclude that A is true, but R is false. Answer: A is true, but R is false.

Question 4 · Linear Algebra MCQ

Assertion (A): In the Gram-Schmidt process, the projection of $v_{2}$ onto $u_{1}$ is calculated as $(v_{2} \cdot u_{1}) u_{1}$ .

Reason (R): This projection vector is orthogonal to $u_{1}$ .

A.
Both A and R are true, and R is the correct explanation of A.
B.
Both A and R are true, but R is not the correct explanation of A.
C.
A is true, but R is false.
D.
A is false, but R is true.

Correct Answer:

Step-by-Step Solution

Key idea: Distinguish between the projection vector and the orthogonalized remainder vector. Step 1: Evaluate Assertion (A). The formula for the projection of

v_{2}

onto the unit vector

u_{1}

is indeed

(v_{2} \cdot u_{1}) u_{1}

. This is correct. Step 2: Evaluate Reason (R). The projection vector

(v_{2} \cdot u_{1}) u_{1}

is a scalar multiple of

u_{1}

, meaning it is parallel to

u_{1}

, not orthogonal to it. Orthogonality is achieved by the remainder vector

w_{2} = v_{2} - (v_{2} \cdot u_{1}) u_{1}

. Step 3: Conclude that A is true, but R is false. Answer: A is true, but R is false.

Question 5 · Linear Algebra MCQ

Let ${u_{1}, u_{2}}$ be an orthonormal basis for $R^{2}$ , where $u_{1} = (0, 1)$ and $u_{2} = (- 1, 0)$ . If a vector $v = (3, 4)$ is expressed as $v = c_{1} u_{1} + c_{2} u_{2}$ , what is the value of the coefficient $c_{2}$ ?

A.
-3
B.
3
C.
4
D.
-4

Correct Answer:

Step-by-Step Solution

Key idea: In an orthonormal basis, coefficients are found using the dot product.

Step 1: Recall the formula for coordinates in an orthonormal basis: $c_{i} = v \cdot u_{i}$ .

Step 2: We need $c_{2}$ , so we compute the dot product of $v$ and $u_{2}$ .

Step 3: $c_{2} = v \cdot u_{2} = (3, 4) \cdot (- 1, 0) = (3) (- 1) + (4) (0) = - 3 + 0 = - 3$ .

Answer: -3

Question 6 · Linear Algebra MCQ

What is the minimum number of vectors in $R^{4}$ that guarantees the set is linearly dependent, regardless of the specific vectors chosen?

A.
3
B.
4
C.
5
D.
6

Correct Answer:

Step-by-Step Solution

Key idea: In $R^{n}$ , any set of more than $n$ vectors is linearly dependent.

Step 1: Identify the dimension of the space. Here, the space is $R^{4}$ , so $n = 4$ .

Step 2: Apply the theorem: Any set of $k$ vectors in $R^{n}$ is linearly dependent if $k > n$ .

Step 3: We need the minimum $k$ such that $k > 4$ . The smallest integer greater than 4 is 5.

Answer: 5

Question 7 · Linear Algebra MCQ

Let ${u_{1}, u_{2}}$ be an orthonormal basis for $R^{2}$ with $u_{1} = (1, 0)$ and $u_{2} = (0, - 1)$ . If a vector $v = (4, - 5)$ is expressed as $v = c_{1} u_{1} + c_{2} u_{2}$ , what is the value of $c_{1} - c_{2}$ ?

A.
-1
B.
9
C.
-9
D.
1

Correct Answer:

Step-by-Step Solution

Key idea: In an orthonormal basis, coordinates are found via dot products.

Step 1: Use the formula $c_{i} = v \cdot u_{i}$ .

Step 2: Calculate $c_{1} = (4, - 5) \cdot (1, 0) = 4 (1) + (- 5) (0) = 4$ .

Step 3: Calculate $c_{2} = (4, - 5) \cdot (0, - 1) = 4 (0) + (- 5) (- 1) = 5$ .

Step 4: Compute the requested value: $c_{1} - c_{2} = 4 - 5 = - 1$ .

Answer: -1

Question 8 · Linear Algebra MCQ

If a matrix $A$ is formed by placing 4 vectors from $R^{6}$ as its columns, what is the minimum rank $A$ must have for these vectors to be linearly independent?

A.
2
B.
4
C.
6
D.
10

Correct Answer:

Step-by-Step Solution

Key idea: The rank method states that vectors are independent if the rank equals the number of vectors.

Step 1: Identify the number of vectors. Here, there are $k = 4$ vectors.

Step 2: Apply the decision rule from the rank method: The vectors are linearly independent if and only if $Rank (A) = k$ .

Step 3: Therefore, the minimum (and exact) rank required is 4. The dimension of the space ( $n = 6$ ) is irrelevant as long as $k \leq n$ .

Answer: 4

Question 9 · Linear Algebra MCQ

Let ${u_{1}, u_{2}}$ be an orthonormal basis for $R^{2}$ , where $u_{1} = (\frac{3}{5}, \frac{4}{5})$ and $u_{2} = (- \frac{4}{5}, \frac{3}{5})$ . If a vector $v = (10, 5)$ is expressed as $v = c_{1} u_{1} + c_{2} u_{2}$ , what is the value of $c_{1} + c_{2}$ ?

A.
5
B.
21
C.
-5
D.
15

Correct Answer:

Step-by-Step Solution

Key idea: In an orthonormal basis, coordinates are found via dot products.

Step 1: Use the formula $c_{i} = v \cdot u_{i}$ .

Step 2: Calculate $c_{1} = (10, 5) \cdot (\frac{3}{5}, \frac{4}{5}) = 10 (\frac{3}{5}) + 5 (\frac{4}{5}) = 6 + 4 = 10$ .

Step 3: Calculate $c_{2} = (10, 5) \cdot (- \frac{4}{5}, \frac{3}{5}) = 10 (- \frac{4}{5}) + 5 (\frac{3}{5}) = - 8 + 3 = - 5$ .

Step 4: Compute the requested sum: $c_{1} + c_{2} = 10 + (- 5) = 5$ .

Answer: 5

Question 10 · Linear Algebra MCQ

If a matrix $A$ is formed by placing 3 vectors from $R^{5}$ as its columns, what is the minimum rank $A$ must have for these vectors to be linearly independent?

A.
2
B.
3
C.
5
D.
8

Correct Answer:

Step-by-Step Solution

Key idea: The rank method states that vectors are independent if the rank equals the number of vectors.

Step 1: Identify the number of vectors. Here, there are $k = 3$ vectors.

Step 2: Apply the decision rule from the rank method: The vectors are linearly independent if and only if $Rank (A) = k$ .

Step 3: Therefore, the minimum (and exact) rank required is 3. The dimension of the space ( $n = 5$ ) is irrelevant as long as $k \leq n$ .

Answer: 3

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Vector Spaces, Subspaces and Bases Practice Questions for GATE DA

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

Vector Spaces, Subspaces and Bases: 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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