GRE Mathematics Test

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​=a(ei−fh)−b(di−fg)+c(dh−eg). Eigenvalues and eigenvectors are tested through characteristic polynomials, such as solving $\det(A - \lambda I) = 0$det(A−λI)=0 for a matrix $A$A, and applications like diagonalization. Inner products, orthogonality, and norms in Euclidean spaces, along with projections, are common, often in $\mathbb{R}^n$Rn. Three-dimensional geometry integrates here via vectors in $\mathbb{R}^3$R3, including lines and planes (e.g., parametric equations for a line through points $P$P and $Q$Q: $\mathbf{r}(t) = \mathbf{P} + t(\mathbf{Q} - \mathbf{P})$r(t)=P+t(Q−P)), dot and cross products ($\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos \theta$u⋅v=∣u∣∣v∣cosθ, $\mathbf{u} \times \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \sin \theta \mathbf{n}$u×v=∣u∣∣v∣sinθn), and orthogonal projections. Quadratic forms, such as classifying conics via $\mathbf{x}^T A \mathbf{x} = c$xTAx=c where $A$A is symmetric, appear in problems linking algebra to geometry. These elements highlight linear independence and transformation properties.^[4] Abstract algebra questions delve into group theory, ring theory, and basic field theory, often requiring proofs of structural properties. Groups are examined through subgroups, cyclic groups (generated by an element $g$g, denoted $\langle g \rangle$⟨g⟩), homomorphisms (structure-preserving maps $\phi: G \to H$ϕ:G→H with $\phi(gh) = \phi(g)\phi(h)$ϕ(gh)=ϕ(g)ϕ(h)), and cosets (left cosets $gH = \{gh \mid h \in H\}$gH={gh∣h∈H} for Lagrange's theorem). Rings cover ideals (subsets closed under addition and multiplication by ring elements) and fields (commutative rings with unity and inverses for non-zero elements). Basic field theory includes extensions (e.g., $\mathbb{Q}(\sqrt{2})$Q(2

​) over $\mathbb{Q}$Q) and finite fields like $\mathbb{F}_p$Fp​ for prime $p$p. Proofs draw from texts such as Dummit and Foote's Abstract Algebra, emphasizing theorems like the first isomorphism theorem. The Fundamental Theorem of Algebra states that every non-constant polynomial with complex coefficients has at least one complex root, though the test focuses on the statement and algebraic implications like polynomial factorization in $\mathbb{C}$C.^[3] Number theory topics include divisibility, primes, congruences, and Diophantine equations, testing modular arithmetic and uniqueness principles. Divisibility properties, such as the Euclidean algorithm for gcd($a, b$a,b) via repeated division, underpin problems like finding integers satisfying $ax + by = d$ax+by=d. Primes and unique factorization (every integer $n > 1$n>1 factors uniquely into primes) support sieve methods or infinitude proofs. Congruences, like $a \equiv b \pmod{m}$a≡b(modm) meaning $m \mid (a - b)$m∣(a−b), lead to Fermat's Little Theorem ($a^{p-1} \equiv 1 \pmod{p}$ap−1≡1(modp) for prime $p \nmid a$p∤a) and Euler's theorem ($a^{\phi(n)} \equiv 1 \pmod{n}$aϕ(n)≡1(modn) where $\phi$ϕ is Euler's totient). Diophantine equations, such as solving $x^2 + y^2 = z^2$x2+y2=z2 for Pythagorean triples, appear in integer solution contexts. These align with elementary number theory from texts like Niven, Zuckerman, and Montgomery's An Introduction to the Theory of Numbers.^[4]

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