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Linear Algebra Short Notes for GATE DA

GATE DA Linear Algebra: 5 units and 23 chapters, weightage from 20 previous year questions across 3 papers, a study order by exam weight and 1055 practice que

A question from this chapter

Question 1

Level 1: Warm-up

Consider the following Assertion (A) and Reason (R) regarding the solution of linear systems with multiple right-hand sides: Assertion (A): When solving A x = b for m different right-hand side vectors, the total computational cost is O (n^{3}) + m \cdot O (n^{2}) . Reason (R): The forward elimination phase must be repeated m times, once for each right-hand side vector, while back substitution is performed only once.

Question 2

Level 1: Warm-up

In Principal Component Analysis (PCA), the first principal component is defined as the direction that:

Question 3

Level 1: Warm-up

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.

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Linear Algebra Short Notes for GATE DA

GATE DA Linear Algebra: 5 units and 23 chapters, weightage from 20 previous year questions across 3 papers, a study order by exam weight and 1055 practice questions.

About Linear Algebra Short Notes

Quick revision sheets for Linear Algebra in GATE DA. Every chapter is condensed into key formulas, shortcuts and common traps so you can revise 23 chapters fast before the exam.

GATE DA Linear Algebra Unit-wise Weightage from Past Papers

We counted every GATE DA Linear Algebra previous year question in our bank (20 questions from 3 papers) and grouped them by unit.

Unit	Chapters	PYQs	Share of section	Avg per paper
Matrices	8	13	65%	4.3
Matrix Decompositions	4	2	10%	0.7
Vector Spaces	5	5	25%	1.7
Unit 1 — Linear Algebra	3	0	0%	0
Unit 2 — Linear Algebra	3	0	0%	0

Suggested Linear Algebra Study Order for GATE DA

Matrices: 65% of past Linear Algebra questions, about 4.3 per paper.
Vector Spaces: 25% of past Linear Algebra questions, about 1.7 per paper.
Matrix Decompositions: 10% of past Linear Algebra questions, about 0.7 per paper.

Start where the marks are. Units at the top of this list have appeared most often in past GATE DA papers.

Units in GATE DA Linear Algebra

All Linear Algebra chapters

One Solved Question from Each Linear Algebra Chapter

Question 1 · Matrix Operations, Determinants and Gaussian Elimination MCQ

Consider the following Assertion (A) and Reason (R) regarding the solution of linear systems with multiple right-hand sides:

Assertion (A): When solving $A x = b$ for $m$ different right-hand side vectors, the total computational cost is $O (n^{3}) + m \cdot O (n^{2})$ .

Reason (R): The forward elimination phase must be repeated $m$ times, once for each right-hand side vector, while back substitution is performed only once.

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: The expensive factorization/elimination step is a property of the matrix $A$ alone, while back substitution depends on the right-hand side $b$ .

Step 1: Forward elimination transforms $A$ into an upper triangular matrix $U$ . This process uses only the entries of $A$ and does not involve $b$ . Therefore, it is performed exactly once, costing $O (n^{3})$ .

Step 2: Back substitution solves $U x = c$ for a specific right-hand side. Since there are $m$ different vectors $b$ , back substitution must be repeated $m$ times, costing $m \cdot O (n^{2})$ .

Step 3: Assertion (A) correctly states the total cost as $O (n^{3}) + m \cdot O (n^{2})$ .

Step 4: Reason (R) incorrectly states that forward elimination is repeated $m$ times and back substitution is performed once. This is the exact opposite of the truth.

Answer: A is true, but R is false.

Question 2 · Singular Value Decomposition and Principal Component Analysis MCQ

In Principal Component Analysis (PCA), the first principal component is defined as the direction that:

A.
Minimizes the reconstruction error of the data
B.
Maximizes the variance of the projected data
C.
Minimizes the trace of the covariance matrix
D.
Maximizes the determinant of the covariance matrix

Correct Answer:

Step-by-Step Solution

Key idea: This is a definition-based question about the primary objective of PCA.

Step 1: Recall that PCA seeks to find orthogonal directions (principal components) that capture the most information in the data.

Step 2: Information in this context is measured by variance. The first principal component is specifically the unit vector $u$ that maximizes the variance of the data when projected onto it.

Answer: Maximizes the variance of the projected data.

Question 3 · Vector Spaces, Subspaces and Bases 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.