GATE DA & AI Engineering Linear Algebra Syllabus

Linear Algebra syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Linear Algebra in GATE DA & AI Engineering, not a summary of it.

1. 1

   Vector space

   2 topics

   • Subspaces
   • Linear dependence and independence of vectors
2. 2

   Matrices

   4 topics

   • Projection matrix
   • Orthogonal matrix
   • Idempotent matrix
   • Partition matrix and their properties
3. 3

   Quadratic forms

   1 topic

   • Systems of linear equations and solutions
     • Gaussian elimination
4. 4

   Eigenvalues and eigenvectors

   overview

   Examined as a single unit within Linear Algebra — no further topic split in the official outline.

5. 5

   Determinant

   overview

   Examined as a single unit within Linear Algebra — no further topic split in the official outline.

6. 6

   Rank and Nullity

   overview

   Examined as a single unit within Linear Algebra — no further topic split in the official outline.

7. 7

   Projections

   overview

   Examined as a single unit within Linear Algebra — no further topic split in the official outline.

8. 8

   LU decomposition

   overview

   Examined as a single unit within Linear Algebra — no further topic split in the official outline.

9. 9

   Singular value decomposition

   overview

   Examined as a single unit within Linear Algebra — no further topic split in the official outline.

Linear Algebra flashcards for GATE DA & AI Engineering

19 of 50 cards from the Linear Algebra deck — real questions with worked answers.

1. What is a subspace of a vector space $V$?

   A non-empty subset $W \subseteq V$ that is itself a vector space under the same operations. Equivalently, $W$ is closed under addition and scalar multiplication, and contains the zero vector $\vec{0}$.

2. State the three conditions a non-empty subset $W$ must satisfy to be a subspace of $V$.

   (1) $\vec{0} \in W$; (2) closure under addition: $\vec{u}, \vec{v} \in W \Rightarrow \vec{u}+\vec{v} \in W$; (3) closure under scalar multiplication: $\vec{v} \in W,\ c \in \mathbb{R} \Rightarrow c\vec{v} \in W$.

3. Give the two trivial subspaces present in every vector space $V$.

   The zero subspace $\{\vec{0}\}$ and the whole space $V$ itself.

4. Is the intersection of two subspaces always a subspace? What about their union?

   The intersection $W_1 \cap W_2$ is always a subspace. The union $W_1 \cup W_2$ is generally NOT a subspace (only when one contains the other).

5. Define the sum $W_1 + W_2$ of two subspaces, and state the dimension formula relating it to the intersection.

   $W_1 + W_2 = \{\vec{w_1}+\vec{w_2} : \vec{w_1}\in W_1,\ \vec{w_2}\in W_2\}$. Dimension formula: $\dim(W_1+W_2) = \dim W_1 + \dim W_2 - \dim(W_1\cap W_2)$.

6. What are the four fundamental subspaces associated with an $m\times n$ matrix $A$?

   Column space $C(A)\subseteq\mathbb{R}^m$, null space $N(A)\subseteq\mathbb{R}^n$, row space $C(A^{T})\subseteq\mathbb{R}^n$, and left null space $N(A^{T})\subseteq\mathbb{R}^m$.

7. When is a set of vectors $\{\vec{v_1},\dots,\vec{v_n}\}$ said to be linearly independent?

   When the only scalars satisfying $c_1\vec{v_1}+c_2\vec{v_2}+\cdots+c_n\vec{v_n}=\vec{0}$ are $c_1=c_2=\cdots=c_n=0$ (the trivial solution).

8. When is a set of vectors linearly dependent?

   When there exist scalars $c_1,\dots,c_n$ not all zero such that $c_1\vec{v_1}+\cdots+c_n\vec{v_n}=\vec{0}$. Equivalently, at least one vector is a linear combination of the others.

9. How can the determinant test be used to check linear independence of $n$ vectors in $\mathbb{R}^{n}$?

   Form the matrix $A$ with the vectors as columns. The vectors are linearly independent if and only if $\det(A)\neq 0$ (equivalently, $A$ is invertible).

10. Relate linear independence of the columns of $A$ to its rank and null space.

    The columns of $A$ ($n$ columns) are linearly independent $\iff \operatorname{rank}(A)=n \iff N(A)=\{\vec{0}\}$ (the homogeneous system $A\vec{x}=\vec{0}$ has only the trivial solution).

11. What can you immediately conclude about linear (in)dependence if a set contains the zero vector?

    Any set of vectors that contains the zero vector $\vec{0}$ is automatically linearly dependent.

12. If you have $k$ vectors in $\mathbb{R}^{n}$ with $k>n$, what is their dependence status?

    They must be linearly dependent. More than $n$ vectors in an $n$-dimensional space can never be independent.

13. Define a basis of a vector space $V$.

    A set of vectors that is both linearly independent and spans $V$. Every vector in $V$ has a unique representation as a linear combination of basis vectors; the number of basis vectors equals $\dim V$.

14. What is the projection of a vector $\vec{b}$ onto a line spanned by vector $\vec{a}$?

    $\operatorname{proj}_{\vec{a}}\vec{b}=\dfrac{\vec{a}^{T}\vec{b}}{\vec{a}^{T}\vec{a}}\,\vec{a}$, with the scalar coefficient $\hat{x}=\dfrac{\vec{a}^{T}\vec{b}}{\vec{a}^{T}\vec{a}}$.

15. Give the projection matrix $P$ that projects any vector onto the line through $\vec{a}$.

    $P=\dfrac{\vec{a}\,\vec{a}^{T}}{\vec{a}^{T}\vec{a}}$ (an $n\times n$ matrix of rank $1$).

16. Give the projection matrix onto the column space of an $m\times n$ matrix $A$ with independent columns.

    $P = A(A^{T}A)^{-1}A^{T}$. It projects any $\vec{b}\in\mathbb{R}^m$ orthogonally onto $C(A)$.

17. State the two defining algebraic properties of any projection matrix $P$.

    It is symmetric and idempotent: $P^{T}=P$ and $P^{2}=P$.

18. If $P$ projects onto subspace $S$, what does $I-P$ project onto?

    $I-P$ projects onto the orthogonal complement $S^{\perp}$. It is also a projection matrix: $(I-P)^{2}=I-P$ and $(I-P)^{T}=I-P$.

19. What are the only possible eigenvalues of a projection matrix, and what do they correspond to?

    The eigenvalues are $0$ and $1$. Eigenvalue $1$ corresponds to vectors already in the subspace (fixed by $P$); eigenvalue $0$ corresponds to vectors in the orthogonal complement (annihilated by $P$).

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