Linear Algebra

Syllabus: Vector space, basis, linear dependence and independence, matrix algebra, eigen values and eigen vectors, rank, solution of linear equations - existence and uniqueness.

9 hand-picked visual resources on this topic, 3 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

Consistency of $Ax=b$

Compare the rank of $A$ with the rank of the augmented matrix $[A|b]$ . If they differ the system is inconsistent — no amount of algebra will produce a solution. If they agree, the gap between that rank and the number of unknowns $n$ counts the free variables.

$\text{rank}(A)\neq\text{rank}([A|b])$
no solution
$\text{rank}(A)=\text{rank}([A|b])=n$

Where marks are lost

A row of $A$ that is a multiple of another makes rank drop — check for that before eliminating.

See it

Eigenvalues and eigenvectors

Solve $\det(A-\lambda I)=0$ . For $2\times2$ and $3\times3$

Matrices with guaranteed spectra

Recognising the type often gives the answer without computation. Symmetry, skew-symmetry and orthogonality each pin down where the eigenvalues can lie.

$A=A^T\Rightarrow \lambda\in\mathbb{R}$
real symmetric: real eigenvalues, orthogonal eigenvectors
$A=-A^T\Rightarrow \lambda \text{ purely imaginary or } 0$

Cayley–Hamilton and similarity

Every matrix satisfies its own characteristic equation, which reduces high powers of $A$ to a low-degree polynomial. Similar matrices ( $B=P^{-1}AP$ ) share eigenvalues; their eigenvectors differ by the factor $P$ .

Basis and independence

$n$ vectors in $\mathbb{R}^n$ form a basis exactly when the matrix they form is non-singular. A determinant set to zero is the standard way GATE asks for the value that breaks independence.

$\det[v_1\ v_2\ \cdots\ v_n]\neq0$

Below: on the syllabus, but no past question for it in this bank.

Projection and least squares
syllabus

Minimising $\lVert e\rVert$ in $v_1=\alpha v_2+e$

Calculus →

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Linear Algebra

Syllabus: Vector space, basis, linear dependence and independence, matrix algebra, eigen values and eigen vectors, rank, solution of linear equations - existence and uniqueness.

9 hand-picked visual resources on this topic, 3 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

Consistency of $Ax=b$

$\text{rank}(A)\neq\text{rank}([A|b])$
no solution
$\text{rank}(A)=\text{rank}([A|b])=n$

Where marks are lost

A row of $A$ that is a multiple of another makes rank drop — check for that before eliminating.

See it

Eigenvalues and eigenvectors

Solve $\det(A-\lambda I)=0$ . For $2\times2$ and $3\times3$

Matrices with guaranteed spectra

Recognising the type often gives the answer without computation. Symmetry, skew-symmetry and orthogonality each pin down where the eigenvalues can lie.

$A=A^T\Rightarrow \lambda\in\mathbb{R}$
real symmetric: real eigenvalues, orthogonal eigenvectors
$A=-A^T\Rightarrow \lambda \text{ purely imaginary or } 0$

Cayley–Hamilton and similarity

Basis and independence

$n$ vectors in $\mathbb{R}^n$ form a basis exactly when the matrix they form is non-singular. A determinant set to zero is the standard way GATE asks for the value that breaks independence.

$\det[v_1\ v_2\ \cdots\ v_n]\neq0$

Below: on the syllabus, but no past question for it in this bank.

Projection and least squares
syllabus

Minimising $\lVert e\rVert$ in $v_1=\alpha v_2+e$

Linear Algebra

Consistency of Ax=bAx=bAx=b

Eigenvalues and eigenvectors

Matrices with guaranteed spectra

Cayley–Hamilton and similarity

Basis and independence

Projection and least squaressyllabus

Linear Algebra

Consistency of Ax=bAx=bAx=b

Eigenvalues and eigenvectors

Matrices with guaranteed spectra

Cayley–Hamilton and similarity

Basis and independence

Projection and least squaressyllabus

Consistency of $Ax=b$

Projection and least squares
syllabus

Consistency of $Ax=b$

Projection and least squares
syllabus