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GATE Mathematics Linear Algebra Flashcards
50 question-and-answer cards covering Linear Algebra as it is examined in GATE Mathematics. 24 of them are printed below, taken from across the deck — no signup, no paywall on the preview.
24 sample cards from the Linear Algebra deck
Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.
If $A$ has characteristic polynomial $(\lambda-2)^{3}$ and minimal polynomial $(\lambda-2)^{2}$, is $A$ diagonalizable?
No. The minimal polynomial has a repeated factor $(\lambda-2)^{2}$, so $A$ is not diagonalizable.
For an eigenvalue $\lambda_i$, how does its exponent in the minimal polynomial relate to Jordan blocks?
The exponent of $(\lambda-\lambda_i)$ in the minimal polynomial equals the size of the largest Jordan block associated with $\lambda_i$.
What is the minimal polynomial of the identity matrix $I_n$ and of a scalar matrix $cI_n$?
For $I_n$: $m(\lambda)=\lambda-1$. For $cI_n$: $m(\lambda)=\lambda-c$. Both are degree $1$, reflecting that scalar matrices are already diagonal.
Why is the minimal polynomial useful for computing the inverse of $A$?
If $m(\lambda)=\lambda^{k}+c_{k-1}\lambda^{k-1}+\cdots+c_1\lambda+c_0$ with $c_0\neq0$, then from $m(A)=0$ one solves $A^{-1}=-\frac{1}{c_0}(A^{k-1}+c_{k-1}A^{k-2}+\cdots+c_1 I)$.
What is an orthonormal set of vectors?
A set $\{u_1,\dots,u_k\}$ is orthonormal if the vectors are mutually orthogonal and each has unit norm: $\langle u_i,u_j\rangle=\delta_{ij}$ (i.e., $0$ for $i\neq j$ and $1$ for $i=j$).
What is the purpose of the Gram-Schmidt orthonormalization process?
It converts a linearly independent set $\{v_1,\dots,v_n\}$ into an orthonormal set $\{u_1,\dots,u_n\}$ spanning the same subspace, producing an orthonormal basis.
Write the Gram-Schmidt formula for the orthogonal vector $w_k$ from $v_k$ given previous orthonormal vectors $u_1,\dots,u_{k-1}$.
$$w_k=v_k-\sum_{j=1}^{k-1}\langle v_k,u_j\rangle\,u_j,\qquad u_k=\frac{w_k}{\lVert w_k\rVert}.$$
In Gram-Schmidt, what is the projection of a vector $v$ onto a nonzero vector $u$?
$$\operatorname{proj}_u(v)=\frac{\langle v,u\rangle}{\langle u,u\rangle}\,u.$$
Apply the first two steps of Gram-Schmidt to $v_1=(1,1,0)$, $v_2=(1,0,1)$ in $\mathbb{R}^3$ (give $u_1$ and the orthogonal $w_2$).
$u_1=\frac{1}{\sqrt{2}}(1,1,0)$. Then $w_2=v_2-\langle v_2,u_1\rangle u_1=(1,0,1)-\frac{1}{2}(1,1,0)=\left(\frac{1}{2},-\frac{1}{2},1\right)$.
What matrix factorization is naturally produced by applying Gram-Schmidt to the columns of $A$?
The QR factorization $A=QR$, where $Q$ has orthonormal columns and $R$ is upper triangular (with the inner products / norms as entries).
Why must Gram-Schmidt be applied to a linearly independent set?
If the vectors are dependent, some $w_k$ becomes the zero vector (since $v_k$ lies in the span of previous vectors), making normalization impossible (division by zero norm).
How does the complex inner product affect the Gram-Schmidt coefficient $\langle v_k,u_j\rangle$?
In the complex case the inner product is conjugate-linear in one slot, so the coefficient $\langle v_k,u_j\rangle$ involves complex conjugation per the chosen convention; otherwise the algorithm is identical to the real case.
Define a unitary matrix $U$.
A complex square matrix $U$ is unitary if $U^{*}U=UU^{*}=I$, where $U^{*}=\overline{U}^{T}$ is the conjugate transpose. Equivalently $U^{-1}=U^{*}$; its columns form an orthonormal set under the complex inner product.
What is the real analogue of a unitary matrix, and its defining property?
An orthogonal matrix $Q$, satisfying $Q^{T}Q=QQ^{T}=I$ (i.e., $Q^{-1}=Q^{T}$); its columns are orthonormal real vectors.
Define a Hermitian matrix and state a key property of its eigenvalues.
A matrix $A$ is Hermitian if $A^{*}=A$ (equal to its conjugate transpose). All eigenvalues of a Hermitian matrix are real, and eigenvectors for distinct eigenvalues are orthogonal.
Define a normal matrix and state the condition it must satisfy.
A matrix $A$ is normal if it commutes with its conjugate transpose: $AA^{*}=A^{*}A$. Hermitian, skew-Hermitian, and unitary matrices are all normal.
State the Spectral Theorem for unitary diagonalization (which matrices are unitarily diagonalizable?).
A complex matrix $A$ is unitarily diagonalizable, i.e., $A=UDU^{*}$ with $U$ unitary and $D$ diagonal, if and only if $A$ is normal ($AA^{*}=A^{*}A$).
Write the formula for diagonalization of a normal matrix by a unitary matrix.
$$A=UDU^{*},$$ where $U$ is unitary ($U^{*}=U^{-1}$), $D=\operatorname{diag}(\lambda_1,\dots,\lambda_n)$, and the columns of $U$ are orthonormal eigenvectors of $A$.
What can be said about the eigenvalues of a unitary matrix?
All eigenvalues $\lambda$ of a unitary matrix have modulus $1$: $|\lambda|=1$, i.e., they lie on the unit circle in $\mathbb{C}$.
What are the eigenvalues of a skew-Hermitian matrix ($A^{*}=-A$)?
They are purely imaginary (of the form $i\beta$ with $\beta\in\mathbb{R}$) or zero.
How is unitary diagonalization connected to the Gram-Schmidt process?
After finding eigenvectors of a normal matrix, Gram-Schmidt is used to orthonormalize them (especially within each eigenspace for repeated eigenvalues), producing the orthonormal columns required for the unitary matrix $U$.
Compare orthogonal diagonalization and unitary diagonalization.
Orthogonal diagonalization $A=QDQ^{T}$ applies to real symmetric matrices (real $D$, orthogonal $Q$). Unitary diagonalization $A=UDU^{*}$ applies to complex normal matrices (possibly complex $D$, unitary $U$). The latter generalizes the former to the complex field.
Why is unitary/orthogonal diagonalization numerically and theoretically preferred over general $PDP^{-1}$ diagonalization?
Because $U^{-1}=U^{*}$ (no separate inverse to compute), unitary matrices preserve norms and inner products (well-conditioned, $\lVert Ux\rVert=\lVert x\rVert$), giving numerical stability and orthonormal eigenbases.
State the determinant relationship for a diagonalizable matrix $A=PDP^{-1}$ and how it gives $\det A$ and $\operatorname{trace} A$.
Determinant and trace are similarity-invariant: $\det A=\det D=\prod_i\lambda_i$ (product of eigenvalues) and $\operatorname{trace} A=\operatorname{trace} D=\sum_i\lambda_i$ (sum of eigenvalues).
What this deck covers
The Linear Algebra deck follows the GATE Mathematics Linear Algebra syllabus — 9 chapters and 5 topics — so questions land on material that is genuinely examinable rather than trivia around it. That works out to roughly 5.6 cards per chapter.
Answers are written to be recallable, not just readable — averaging about 160 characters, which is long enough to carry the reasoning and short enough to say out loud.
A deck like this earns its keep on the second and third pass. Read the syllabus first so you know the shape of the subject, then use the cards to find the specific facts that have not stuck.
Linear Algebra flashcards FAQ
How many Linear Algebra flashcards are in this GATE Mathematics deck?
50 cards. This page previews 24 of them, sampled evenly across the deck so you can judge the difficulty before installing anything.
Are these GATE Mathematics flashcards free?
Yes. The preview here is free to read with no signup, and the full 50-card deck is free inside the Examius app.
What do the Linear Algebra cards cover?
They follow the GATE Mathematics Linear Algebra syllabus — 9 chapters and 5 topics — so the questions track what is actually examinable.
How should I use these flashcards?
Read the syllabus first so you know the shape of the subject, then drill the deck. Examius schedules each card with spaced repetition, so cards you keep missing come back sooner and ones you know drift further apart.