GATE Mathematics 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 Mathematics, not a summary of it.

1. 1

   Finite Dimensional Vector Spaces

   1 topic

   • Real or Complex Fields
2. 2

   Linear Transformations and Matrix Representations

   1 topic

   • Rank and Nullity
3. 3

   Systems of Linear Equations

   overview

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

4. 4

   Characteristic Polynomial, Eigen Values, and Eigen Vectors

   1 topic

   • Diagonalization
     • Minimal Polynomial
5. 5

   Cayley-Hamilton Theorem

   overview

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

6. 6

   Finite Dimensional Inner Product Spaces

   1 topic

   • Gram-Schmidt Orthonormalization Process
7. 7

   Symmetric, Skew-Symmetric, Hermitian, Skew-Hermitian, Normal, Orthogonal, and Unitary Matrices

   1 topic

   • Diagonalization by a Unitary Matrix
8. 8

   Jordan Canonical Form

   overview

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

9. 9

   Bilinear and Quadratic Forms

   overview

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

Linear Algebra flashcards for GATE Mathematics

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

1. What is a field $F$ in linear algebra, and which two fields are most commonly used as scalar fields?

   A field is a set with two operations (addition and multiplication) satisfying commutativity, associativity, distributivity, existence of additive and multiplicative identities, and inverses (multiplicative inverse for every nonzero element). The two standard scalar fields are the real field $\mathbb{R}$ and the complex field $\mathbb{C}$.

2. How does $\mathbb{C}$ differ from $\mathbb{R}$ regarding algebraic closure, and why does this matter for eigenvalues?

   $\mathbb{C}$ is algebraically closed: every non-constant polynomial has all roots in $\mathbb{C}$. $\mathbb{R}$ is not. Hence every $n\times n$ matrix has $n$ eigenvalues (with multiplicity) over $\mathbb{C}$, but a real matrix may have no real eigenvalues.

3. What is the dimension of $\mathbb{C}$ as a vector space over $\mathbb{R}$, and over $\mathbb{C}$?

   As a vector space over $\mathbb{R}$, $\dim_{\mathbb{R}}\mathbb{C}=2$ (basis $\{1,i\}$). As a vector space over itself, $\dim_{\mathbb{C}}\mathbb{C}=1$.

4. For a complex inner product space, how is the inner product $\langle x,y\rangle$ different from the real case in terms of symmetry?

   It is conjugate-symmetric (Hermitian): $\langle x,y\rangle=\overline{\langle y,x\rangle}$, and it is conjugate-linear in one argument. In the real case it is simply symmetric: $\langle x,y\rangle=\langle y,x\rangle$.

5. Define the rank of a matrix $A$.

   The rank of $A$ is the dimension of its column space (equivalently its row space): the maximum number of linearly independent columns (or rows). It equals the number of nonzero rows in the row-echelon form, or the order of the largest nonzero minor.

6. Define the nullity of an $m\times n$ matrix $A$.

   The nullity of $A$ is $\dim(\ker A)$, the dimension of the null space $\{x\in F^{n}: Ax=0\}$, i.e., the number of free variables / independent solutions of $Ax=0$.

7. State the Rank-Nullity Theorem for an $m\times n$ matrix $A$.

   $$\operatorname{rank}(A)+\operatorname{nullity}(A)=n,$$ where $n$ is the number of columns (the dimension of the domain).

8. For a linear transformation $T:V\to W$, state the Rank-Nullity (dimension) theorem.

   $$\dim(\operatorname{Im} T)+\dim(\ker T)=\dim V,$$ i.e., $\operatorname{rank}(T)+\operatorname{nullity}(T)=\dim V$.

9. How are the rank of $A$ and the rank of $A^{T}$ related?

   They are equal: $\operatorname{rank}(A)=\operatorname{rank}(A^{T})$. This expresses that row rank equals column rank.

10. What does it mean for a square $n\times n$ matrix $A$ to be of full rank, and what does it imply about invertibility and nullity?

    Full rank means $\operatorname{rank}(A)=n$. Then $A$ is invertible, $\det A\neq 0$, $\operatorname{nullity}(A)=0$, and $Ax=0$ has only the trivial solution.

11. State the rank inequality for the product of matrices $A$ and $B$ (Sylvester's rank inequality form).

    $\operatorname{rank}(AB)\le\min\{\operatorname{rank}(A),\operatorname{rank}(B)\}$. Sylvester's inequality: $\operatorname{rank}(A)+\operatorname{rank}(B)-n\le\operatorname{rank}(AB)$, where $A$ is $m\times n$ and $B$ is $n\times p$.

12. For the system $Ax=b$ with $A$ being $m\times n$, what consistency condition involves rank?

    The system is consistent (has a solution) if and only if $\operatorname{rank}(A)=\operatorname{rank}([A\,|\,b])$, the rank of the augmented matrix equals the rank of the coefficient matrix.

13. For a consistent system $Ax=b$ with $A$ of size $m\times n$ and rank $r$, how many free parameters does the solution set have?

    It has $n-r$ free parameters. If $r=n$ the solution is unique; if $r<n$ there are infinitely many solutions forming an affine subspace of dimension $n-r$.

14. Define an eigenvalue and eigenvector of a square matrix $A$.

    A scalar $\lambda$ is an eigenvalue with nonzero eigenvector $x$ if $Ax=\lambda x$, equivalently $(A-\lambda I)x=0$ has a nontrivial solution, i.e., $\det(A-\lambda I)=0$.

15. What is the characteristic polynomial of an $n\times n$ matrix $A$?

    $$p(\lambda)=\det(A-\lambda I),$$ a degree-$n$ polynomial whose roots are the eigenvalues of $A$.

16. Distinguish algebraic multiplicity and geometric multiplicity of an eigenvalue $\lambda$.

    Algebraic multiplicity is the multiplicity of $\lambda$ as a root of the characteristic polynomial. Geometric multiplicity is $\dim\ker(A-\lambda I)$, the number of independent eigenvectors for $\lambda$. Always $1\le \text{geometric}\le \text{algebraic}$.

17. What is the necessary and sufficient condition for an $n\times n$ matrix $A$ to be diagonalizable?

    $A$ is diagonalizable iff it has $n$ linearly independent eigenvectors, equivalently iff for every eigenvalue the geometric multiplicity equals the algebraic multiplicity (the sum of geometric multiplicities is $n$).

18. If $A$ is diagonalizable, write the diagonalization formula.

    $A=PDP^{-1}$, where $D=\operatorname{diag}(\lambda_1,\dots,\lambda_n)$ holds the eigenvalues and the columns of $P$ are the corresponding linearly independent eigenvectors.

19. Give a sufficient (not necessary) condition for an $n\times n$ matrix to be diagonalizable based on eigenvalues.

    If $A$ has $n$ distinct eigenvalues, then it is diagonalizable. (This is sufficient but not necessary; matrices with repeated eigenvalues can still be diagonalizable.)

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