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Analysis & Linear Algebra

Inner Product Spaces and Forms

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1. Gram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem

Read this as a chapter →

Lec-40 Norms on Vector spaces. The Gram-Schmidt Procedure I

NPTEL · Linear Algebra

Lec-50 Self-Adjoint Operators II - Spectral Theorem

NPTEL · Linear Algebra

Spectral theorem for self-adjoint operators.

Lec-51 Normal Operators - Spectral Theorem

NPTEL · Linear Algebra

Normality is exactly the condition for unitary diagonalisability.

Inner products, orthogonality and the spectral theorem free

In plain words

Gram–Schmidt straightens a set of directions into perpendicular ones. The spectral theorem says a symmetric matrix diagonalises using eigenvectors at right angles, and that is what makes it a tool.

Inner products

⟨·,·⟩ must be linear (in one slot), conjugate-symmetric and positive definite. Testing candidates: ∑jwjxjyj\sum_j w_j x_j y_j is an inner product ⇔ all w_j > 0. Forms like ∑i,jxiyj\sum_{i,j}x_iy_j =(∑x)(∑y)= (\sum{}x)(\sum{}y) are degenerate, and ∑xj\sum x_j yn−j+1y_{n-j+1} is symmetric but indefinite.

Cauchy–Schwarz |⟨x,y⟩| ≤ ‖x‖‖y‖; parallelogram law characterises norms coming from an inner product.

Gram–Schmidt

Produces an orthonormal basis; gives the QR decomposition A = QR. The Gram matrix G = (⟨vi,vjv_{i},v_{j}⟩) is positive semi-definite, and positive definite ⇔ the viv_{i} are independent.

The matrix classes

Class Definition Diagonalisable?
symmetric (real) Aᵀ = A ✓ orthogonally, real eigenvalues
Hermitian A* = A ✓ unitarily, real eigenvalues
skew-symmetric (real) Aᵀ = −A ✓ over C\mathbb{C}; eigenvalues purely imaginary
orthogonal AᵀA = I ✓ over C\mathbb{C}; |λ\lambda| = 1
unitary A*A = I ✓ unitarily; |λ\lambda| = 1
normal AA = AA ✓ unitarily (spectral theorem)

All the above are normal — normality is exactly the condition for unitary diagonalisability over C\mathbb{C}.

Spectral theorem consequences

  • Real symmetric: eigenvectors for distinct eigenvalues are orthogonal; A=QΛQTA = Q\Lambda{}Q^{T}.
  • A symmetric with Ak=I⇒A2=I(A^{k} = I \Rightarrow A^{2} = I (real eigenvalues with λk=1⇒λ=±1)\lambda^{k} = 1 \Rightarrow \lambda = \pm1).
  • A symmetric with Aᵏ = 0 ⇒ A = 0.
  • Positive definite ⇔ all eigenvalues > 0 ⇔ all leading principal minors > 0 ⇔ A = BᵀB with B invertible.

Orthogonal projections

P is an orthogonal projection ⇔P2=P\Leftrightarrow P^{2} = P and P = Pᵀ. Then Rn=\mathbb{R}^{n} = im P ⊕ ker P orthogonally, and rank P = trace P.

Key takeaways

  • Normal ⇔ unitarily diagonalisable; symmetric/Hermitian/unitary/skew are all special cases.
  • Real symmetric gives a real orthonormal eigenbasis — the strongest statement.
  • Positive weights are what make ∑wjxjyj\sum{}w_{j}x_{j}y_{j} an inner product.

Gram–Schmidt, one projection at a timeinteractive

This visual is part of Trap Analysis

Steer v₂ and watch its shadow on e₁ get subtracted — and see the algorithm genuinely halt when the vectors go dependent.

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Exam focus: Spectral theorem: real symmetric ⇒ orthogonally diagonalisable with real eigenvalues; complex normal ⇒ unitarily diagonalisable. Know which matrix classes are normal.

Practice

Part B2025 Juneshow ▾
Let C[0,π]C[0, \pi] be the real vector space of real-valued continuous functions on the closed interval [0,π][0, \pi]. For positive integers n, define fn∈C[0,π]f_{n} \in C[0, \pi] by fn(x)=sin⁡(f_{n}(x) = \sin(nx)/sin x if x∈(0,π),fn(0)=nx \in (0, \pi), f_{n}(0) = n, and fn(π)=(−1)nf_{n}(\pi) = (-1)^{n}⁻1n^{1}n. Let V be the real subspace of C[0,π]C[0, \pi] spanned by {f1,f2,f3f_{1}, f_{2}, f_{3}}. Consider the inner product on V given by ⟨f, g⟩ =(2/π)∫0πf(x)g(x)sin⁡2x= (2/\pi)\int_{0}^\pi f(x)g(x)\sin^{2}x dx. Which of the following statements is true?
  1. A.f4∈Vf_{4} \in V
  2. B.{f1,f2,f3f_{1}, f_{2}, f_{3}} is an orthonormal basis of V.✓ Correct
  3. C.The dimension of V is 2.
  4. D.{f1,f2,f3f_{1}, f_{2}, f_{3}} is an orthogonal set but not orthonormal.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Decembershow ▾
For n≥3, consider the space Mn(C)M_{n}(\mathbb{C}) of n×n complex matrices endowed with the inner product ⟨A,B⟩ = Trace(A*B). For 0≤k≤n, let Wk=W_{k} = {A∈Mn(C)A \in M_{n}(\mathbb{C}) : ⟨A,B⟩=0 for all B∈Mn(C)B \in M_{n}(\mathbb{C}) with rank k}. Which of the following statements are necessarily true?
  1. A.W0=W_{0} = {0}
  2. B.W1=W_{1} = {0}✓ Correct
  3. C.W2=W_{2} = {0}✓ Correct
  4. D.Wn=W_{n} = {0}✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Decembershow ▾
Let B be a 4×4 positive-definite real symmetric matrix which is not the identity matrix. Consider the inner product on R4\mathbb{R}^{4} given by ⟨v,w⟩ = vᵀBw, where vᵀ denotes the transpose of v. Which of the following statements is FALSE?
  1. A.If v and w are eigenvectors of B for distinct eigenvalues, then ⟨v,w⟩ = 0.
  2. B.If v is an eigenvector of B and w ≠ 0 is such that ⟨w,v⟩ = 0, then w is an eigenvector of B.✓ Correct
  3. C.For every subspace W⊆R4,W+W⊥=R4W \subseteq \mathbb{R}^{4}, W + W\perp = \mathbb{R}^{4}, where W⊥ = {v∈R4v \in \mathbb{R}^{4} ∣ ⟨v,w⟩ = 0 for all w ∈ W}.
  4. D.If W⊆R4W \subseteq \mathbb{R}^{4} is an eigenspace of B, then W has an orthonormal basis.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾
Let V be a finite dimensional complex inner product space. For a linear map T : V → V, let T* denote its adjoint. Which of the following statements are true?
  1. A.If trace of TT* is zero, then T = 0.✓ Correct
  2. B.Let v ∈ V be such that T*T(v) = 0. Then T(v) = 0.✓ Correct
  3. C.Suppose T = T* and N > 1 be an integer. Let v ∈ V be such that T2N(v)T^{2^N}(v) = 0. Then T(v) = 0.✓ Correct
  4. D.Suppose that TT* = T*T and N > 1 be an integer. Let v ∈ V be such that T^N(v) = 0. Then T(v) = 0.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let (V, ⟨ , ⟩) be an inner product space over C\mathbb{C} and T : V → V be aC−a \mathbb{C}-linear transformation. Let v1v_{1} and v2v_{2} be non-zero vectors in V such that Tv1=c1v1_{1} = c_{1}v_{1} and Tv2=c2v2_{2} = c_{2}v_{2} for some c1,c2∈Cc_{1}, c_{2} \in \mathbb{C}. Let v3=v2−(v_{3} = v_{2} - (⟨v2,v1v_{2}, v_{1}⟩/‖v1v_{1}‖2)v1^{2})v_{1}. Suppose that v3v_{3} is non-zero and Tv3=c3v3_{3} = c_{3}v_{3} for some c3∈Cc_{3} \in \mathbb{C}. Which of the following statements are true?
  1. A.The set {v1,v2v_{1}, v_{2}} is linearly independent.✓ Correct
  2. B.If c1≠c2c_{1} \ne c_{2}, then ⟨v1,v2v_{1}, v_{2}⟩ = 0.✓ Correct
  3. C.If c1=c2c_{1} = c_{2}, then c3=c2c_{3} = c_{2}.✓ Correct
  4. D.If c1≠c2c_{1} \ne c_{2}, then c3=c2c_{3} = c_{2}.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾
Consider R4\mathbb{R}^{4} with the standard inner product. Let V be the subspace of R4\mathbb{R}^{4} spanned by the vectors (1, 0, 0, 1), (0, 1, 0, 1), and (0, 0, 1, 0). Which of the following is NOT an orthonormal basis of V?
  1. A.{(−1/2,0,0,−1/2),(−1/6,2/3,0,1/6),(0,0,1,0)(-1/\sqrt{2}, 0, 0, -1/\sqrt{2}), (-1/\sqrt{6}, \sqrt{2}/\sqrt{3}, 0, 1/\sqrt{6}), (0, 0, 1, 0)}
  2. B.{(1/2,0,0,1/2),(−1/6,2/3,0,1/6),(0,0,1,0)(1/\sqrt{2}, 0, 0, 1/\sqrt{2}), (-1/\sqrt{6}, \sqrt{2}/\sqrt{3}, 0, 1/\sqrt{6}), (0, 0, 1, 0)}
  3. C.{(1/2,0,0,1/2),(−1/6,2/3,0,−1/6),(0,0,1,0)(1/\sqrt{2}, 0, 0, 1/\sqrt{2}), (-1/\sqrt{6}, \sqrt{2}/\sqrt{3}, 0, -1/\sqrt{6}), (0, 0, 1, 0)}✓ Correct
  4. D.{(1/2,0,0,1/2),(1/6,−2/3,0,−1/6),(0,0,1,0)(1/\sqrt{2}, 0, 0, 1/\sqrt{2}), (1/\sqrt{6}, -\sqrt{2}/\sqrt{3}, 0, -1/\sqrt{6}), (0, 0, 1, 0)}

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let V be the R−\mathbb{R}-vector space of real valued continuous functions on the interval [0,π][0, \pi] with the inner product given by ⟨f, g⟩ =∫0πf(x)g(x)= \int_{0}^\pi f(x)g(x)dx. Let S = {sin⁡(x),cos⁡(x),sin⁡2(x),cos⁡2(x)\sin(x), \cos(x), \sin^{2}(x), \cos^{2}(x)} and W be the subspace of V generated by S. Which of the following statements are true?
  1. A.S is a basis of W.✓ Correct
  2. B.S is an orthonormal basis of W.
  3. C.There exist f, g ∈ S such that ⟨f, g⟩ = 0.✓ Correct
  4. D.S contains an orthonormal basis of W.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾
Consider the real vector space V=R[x]V = \mathbb{R}[x] equipped with an inner product. Let W be the subspace of V consisting of polynomials of degree at most 2. Let W^⊥ denote the orthogonal complement of W in V. Which of the following statements are true?
  1. A.There exists a polynomial p(x) ∈ W such that x4−p(x)∈W⊥x^{4} - p(x) \in W^\perp✓ Correct
  2. B.W^⊥ = {0}
  3. C.W and W^⊥ have the same dimension over R\mathbb{R}
  4. D.W^⊥ is an infinite dimensional vector space over R\mathbb{R}✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Juneshow ▾
Let x=(x1,…,xn)x = (x_{1}, \dots, x_{n}) and y=(y1,…,yn)y = (y_{1}, \dots, y_{n}) denote vectors in Rn\mathbb{R}^{n} for a fixed n ≥ 2. Which of the following defines an inner product on Rn\mathbb{R}^{n}?
  1. A.⟨x, y⟩ =∑i,j=1n= \sum_{i,j=1}^n xiyjx_{i}y_{j}
  2. B.⟨x, y⟩ =∑i,j=1n= \sum_{i,j=1}^n (xi2+yj2)(x_{i}^{2} + y_{j}^{2})
  3. C.⟨x, y⟩ =∑j=1n= \sum_{j=1}^n j3xjyjj^{3} x_{j}y_{j}✓ Correct
  4. D.⟨x, y⟩ =∑j=1n= \sum_{j=1}^n xjynx_{j} y_{n}₋j_{j}₊1_{1}

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾
Let U be a real orthogonal n × n matrix (UᵀU = I). Which of the following are true?
  1. A.det U = ±1.✓ Correct
  2. B.Every real eigenvalue of U is 1 or −1.✓ Correct
  3. C.U preserves the Euclidean norm: ‖Ux‖ = ‖x‖ for all x.✓ Correct
  4. D.U is necessarily symmetric.

Solution

(1)det⁡(UTU)=(det⁡U)2=1(1) \det(U^{T}U) = (\det U)^{2} = 1.

(2) If Ux =λx= \lambda{}x with real x ≠ 0, then ‖x‖ = ‖Ux‖ = |λ\lambda|‖x‖ forces |λ\lambda| = 1.

(3) ‖Ux‖2=xTUT^{2} = x^{T}U^{T}Ux = ‖x‖2^{2}.

(4) fails: the 90° rotation [[0, −1], [1, 0]] is orthogonal and not symmetric.

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2. Quadratic forms, positive definiteness, Sylvester's law

Read this as a chapter →

Positive semi-definite matrix, monotonicity theorem and interlacing

NPTEL · Matrix Theory

Definiteness via eigenvalues; pair it with the leading-principal-minor test in the notes.

Inner product

NPTEL · Matrix Theory

Positive definiteness of a form is the same condition that makes an inner product.

Quadratic forms, signature and definiteness free

In plain words

Only the counts of plus signs, minus signs and zeros survive a change of variables, and that triple classifies the form completely. Positive definite means every direction gives a positive value.

Sylvester's law of inertia

Every real quadratic form is congruent to a diagonal form with p entries +1, q entries −1 and the rest 0. The signature (p, q) is an invariant: two real quadratic forms are equivalent (congruent) ⇔ same rank and signature.

Over C\mathbb{C} only the rank matters.

Definiteness tests

Test Criterion for positive definite
Eigenvalues all λ>0\lambda > 0
Leading principal minors all > 0 (Sylvester's criterion)
All principal minors needed for positive semi-definite (leading minors are not enough)
Congruence A = BᵀB with B invertible

The trap: det > 0 does not give positive definite — diag(−1,−1) has det 1. You need all leading principal minors positive, in order.

Worked: A_a = [[2,−1,0],[−1,2,−1],[0,−1,a]] has minors 2, 3, 3a − 2, so it is positive definite ⇔ a > 2/3.

Binary forms

ax2+^{2} + bxy + cy2^{2}: discriminant b2−4b^{2} - 4ac.

  • < 0 with a > 0: positive definite;
  • < 0 with a < 0: negative definite;

  • 0: indefinite (signature (1,1));

  • = 0: degenerate.

Two non-degenerate binary forms over R\mathbb{R} are equivalent ⇔ same signature. So 6X2−136X^{2} - 13XY +6Y2(+ 6Y^{2} (disc 25 > 0) and X2−X^{2} - XY −2Y2(- 2Y^{2} (disc 9 > 0) are equivalent, while X2−X^{2} - XY +2Y2(+ 2Y^{2} (disc −7) is positive definite and equivalent to neither.

Indefinite ⇒ isotropic vectors

An indefinite non-degenerate form has non-zero v with ⟨v,v⟩ = 0: take v=u/√v = u/\surd⟨u,u⟩ +w/√(−+ w/\surd(-⟨w,w⟩) for ⟨u,u⟩ > 0 > ⟨w,w⟩.

Key takeaways

  • Signature is the complete real invariant; rank alone is the complex one.
  • Positive definite ⇔ all leading principal minors positive.
  • Indefinite forms always have isotropic vectors.

Testing definiteness, and reading Sylvester's lawinteractive

This visual is part of Trap Analysis

The leading-minors test done right — including the sign pattern for negative definite that catches people out.

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Exam focus: Signature is the complete invariant over ℝ (Sylvester). Positive definiteness is tested by leading principal minors, or by all eigenvalues being positive — do not mix the two tests up.

Practice

Part B2025 Decembershow ▾
Consider the quadratic form f(x,y,z) = [x y z] A [x;y;z], where A is an invertible 3×3 symmetric matrix over Q\mathbb{Q}. Assume that there exists (α,β,γ)∈C3(\alpha,\beta,\gamma) \in \mathbb{C}^{3}{(0,0,0)} such that f(α,β,γ)=0f(\alpha,\beta,\gamma) = 0. Which of the following statements is necessarily true?
  1. A.There exists (a,b,c)∈R3(a,b,c) \in \mathbb{R}^{3}{(0,0,0)} such that f(a,b,c) = 0.
  2. B.If there exists (a,b,c)∈R3(a,b,c) \in \mathbb{R}^{3}{(0,0,0)} such that f(a,b,c) = 0, then (a,b,c)∈Q3(a,b,c) \in \mathbb{Q}^{3}.
  3. C.{(a,b,c)∈Z3(a,b,c) \in \mathbb{Z}^{3} ∣ f(a,b,c) = 0} is a finite set.
  4. D.If there exists (a,b,c)∈Q3(a,b,c) \in \mathbb{Q}^{3}{(0,0,0)} such that f(a,b,c) = 0, then A has a positive eigenvalue and a negative eigenvalue.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾
Let B(v, w) be a nondegenerate symmetric bilinear form on R2\mathbb{R}^{2} and let q(v) = B(v, v) be the corresponding quadratic form. Suppose there exist vectors v,w∈R2v, w \in \mathbb{R}^{2} such that B(v, v) = 0 and B(v, w) ≠ 0. Which of the following statements are necessarily true?
  1. A.B(w, w) = 0
  2. B.There exists an α∈R\alpha \in \mathbb{R} such that q(αv+w)=0q(\alpha{}v + w) = 0.✓ Correct
  3. C.There are infinitely many α∈R\alpha \in \mathbb{R} such that q(αv+w)=0q(\alpha{}v + w) = 0.
  4. D.q is equivalent to the quadratic form Q(x,y)=x2−y2Q(x, y) = x^{2} - y^{2} for all (x,y)∈R2(x, y) \in \mathbb{R}^{2}.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2025 Juneshow ▾
Let V = {ax3+^{3} + bx2+^{2} + cx | a,b,c∈Ra, b, c \in \mathbb{R}}. For f ∈ V, define Q(f)=∫Q(f) = \int₋11(f′(t))2_{1}^{1} (f'(t))^{2} dt, where f′ denotes the derivative of f. Which of the following statements is FALSE?
  1. A.Q is a positive definite quadratic form on V.
  2. B.Q takes every positive real value.
  3. C.Q(x) = 2.
  4. D.For all f, g ∈ V, Q(f + g) = Q(f) + Q(g).✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾
Consider the real matrix A = [[0, 0, 1], [0, 1, 0], [1, 0, 0]]. Define B:R3×R3→RB : \mathbb{R}^{3} \times \mathbb{R}^{3} \to \mathbb{R} by B(v, w) = vᵗAw. Which of the following statements is true?
  1. A.B(v, v) = 0 if and only if v = 0.
  2. B.For every λ∈R\lambda \in \mathbb{R}, there exists v such that B(v,v)=λB(v, v) = \lambda.✓ Correct
  3. C.There exists v ≠ 0 such that B(v, w) = 0 for all w∈R3w \in \mathbb{R}^{3}.
  4. D.If B(v, w) = 0 then either v = 0 or w = 0.

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Decembershow ▾
Consider the bilinear form B:R4×R4→RB : \mathbb{R}^{4} \times \mathbb{R}^{4} \to \mathbb{R} defined by B(x,y)=x1y3+x2y4−x3y1−x4y2B(x, y) = x_{1}y_{3} + x_{2}y_{4} - x_{3}y_{1} - x_{4}y_{2}, where x=(x1,x2,x3,x4)x = (x_{1}, x_{2}, x_{3}, x_{4}) and y=(y1,y2,y3,y4)y = (y_{1}, y_{2}, y_{3}, y_{4}) in R4\mathbb{R}^{4}. Let A denote the matrix of B with respect to the standard ordered basis of R4\mathbb{R}^{4}. Which of the following statements is true?
  1. A.det A = 0
  2. B.det A = −1
  3. C.B(x, x) ≠ 0 for all nonzero x∈R4x \in \mathbb{R}^{4}.
  4. D.If x∈R4x \in \mathbb{R}^{4} is nonzero, then there exists y∈R4y \in \mathbb{R}^{4} such that B(x, y) ≠ 0.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
For a 4 × 4 positive definite real symmetric matrix A and real numbers a, b, c, d, consider the 5 × 5 matrix B whose first row is (0, a, b, c, d), whose first column is (0, a, b, c, d)ᵗ, and whose lower-right 4 × 4 block is A. Which of the following statements are necessarily true?
  1. A.det(B) > 0 for every nonzero (a,b,c,d)∈R4(a, b, c, d) \in \mathbb{R}^{4}.
  2. B.det(B) > 0 for infinitely many (a,b,c,d)∈R4(a, b, c, d) \in \mathbb{R}^{4}.
  3. C.det(B) ≤ 0 for every (a,b,c,d)∈R4(a, b, c, d) \in \mathbb{R}^{4}.✓ Correct
  4. D.det(B) ≤ 0 for infinitely many (a,b,c,d)∈R4(a, b, c, d) \in \mathbb{R}^{4}.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾
Let A = ((0, 1, 0, 0), (1, 0, 0, 0), (0, 0, 1, 1), (0, 0, 1, 1)), and consider the symmetric bilinear form on R4\mathbb{R}^{4} given by ⟨v, w⟩ = vᵗAw, for v,w∈R4v, w \in \mathbb{R}^{4}. Which of the following statements is true?
  1. A.A is invertible
  2. B.There exist non-zero vectors v, w such that ⟨v, w⟩ = 0✓ Correct
  3. C.⟨u, v⟩ ≠ ⟨u, w⟩ for all non-zero vectors u, v, w with v ≠ w
  4. D.Every eigenvalue of A2A^{2} is positive

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let V be a two-dimensional real vector space with a basis {v1,v2v_{1}, v_{2}}. Let f:V×V→Rf : V \times V \to \mathbb{R} be a symmetric bilinear form. Let r=f(v1,v1),s=f(v1,v2),t=f(v2,v2)r = f(v_{1}, v_{1}), s = f(v_{1}, v_{2}), t = f(v_{2}, v_{2}). Which of the following statements are necessarily true?
  1. A.f(v2,v1)=−sf(v_{2}, v_{1}) = -s.
  2. B.If r = s = t, then either f(v, v) ≥ 0 for all v ∈ V or f(v, v) ≤ 0 for all v ∈ V.✓ Correct
  3. C.If f is positive definite, then r ≠ s, s ≠ t and r ≠ t.
  4. D.If f is positive definite, then f(v1,v2)=0f(v_{1}, v_{2}) = 0.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾
Let q1(x1,x2)q_{1}(x_{1}, x_{2}) and q2(y1,y2)q_{2}(y_{1}, y_{2}) be real quadratic forms such that there exist (u1,u2),(v1,v2)∈R2(u_{1}, u_{2}), (v_{1}, v_{2}) \in \mathbb{R}^{2} such that q1(u1,u2)=1=q2(v1,v2)q_{1}(u_{1}, u_{2}) = 1 = q_{2}(v_{1}, v_{2}). Define q(x1,x2,y1,y2)=q1(x1,x2)−q2(y1,y2)q(x_{1}, x_{2}, y_{1}, y_{2}) = q_{1}(x_{1}, x_{2}) - q_{2}(y_{1}, y_{2}). Which of the following statements are necessarily true?
  1. A.q is a quadratic form in x1,x2,y1,y2x_{1}, x_{2}, y_{1}, y_{2}✓ Correct
  2. B.There exists (t1,t2)∈R2(t_{1}, t_{2}) \in \mathbb{R}^{2} such that q1(t1,t2)=5q_{1}(t_{1}, t_{2}) = 5✓ Correct
  3. C.There does not exist (s1,s2)∈R2(s_{1}, s_{2}) \in \mathbb{R}^{2} such that q2(s1,s2)=−5q_{2}(s_{1}, s_{2}) = -5
  4. D.Given α∈R\alpha \in \mathbb{R}, there exists a vector w∈R4w \in \mathbb{R}^{4} such that q(w)=αq(w) = \alpha✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2024 Juneshow ▾
For a quadratic form f(x,y,z)∈R[x,y,z]f(x, y, z) \in \mathbb{R}[x, y, z], we say that (a,b,c)∈R3(a, b, c) \in \mathbb{R}^{3} is a zero of f if f(a, b, c) = 0. Which of the following quadratic forms has at least one zero different from (0, 0, 0)?
  1. A.x2+2y2+3z2x^{2} + 2y^{2} + 3z^{2}
  2. B.x2+2y2+3z2−2x^{2} + 2y^{2} + 3z^{2} - 2xy
  3. C.x2+2y2+3z2−2x^{2} + 2y^{2} + 3z^{2} - 2xy − 2yz
  4. D.x2+2y2−3z2x^{2} + 2y^{2} - 3z^{2}✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾
Let (−, −) be a symmetric bilinear form on R2\mathbb{R}^{2} such that there exist nonzero v,w∈R2v, w \in \mathbb{R}^{2} with (v, v) > 0 > (w, w) and (v, w) = 0. Let A be the 2×2 real symmetric matrix representing this form in the standard basis. Which one of the following statements is true?
  1. A.A2=0A^{2} = 0.
  2. B.rank A = 1.
  3. C.rank A = 0.
  4. D.There exists u∈R2,u≠0u \in \mathbb{R}^{2}, u \ne 0 such that (u, u) = 0.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part B2023 Decembershow ▾
For a∈Ra \in \mathbb{R}, let A_a = [[2, −1, 0], [−1, 2, −1], [0, −1, a]]. Which one of the following statements is true?
  1. A.A_a is positive definite for all a < 3.
  2. B.A_a is positive definite for all a > 3.✓ Correct
  3. C.A_a is positive definite for all a ≥ −2.
  4. D.A_a is positive definite only for finitely many values of a.

Solution

Leading principal minors: 2, 3 and det A_a = 3a − 2.

Positive definite iff a > 2/3, which contains all a > 3.

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Part C2023 Juneshow ▾
Consider the following quadratic forms over R:(a)6X2−13\mathbb{R}: (a) 6X^{2} - 13XY +6Y2,(b)X2−+ 6Y^{2}, (b) X^{2} - XY +2Y2,(c)X2−+ 2Y^{2}, (c) X^{2} - XY −2Y2- 2Y^{2}. Which of the following statements are true?
  1. A.Quadratic forms (a) and (b) are equivalent.
  2. B.Quadratic forms (a) and (c) are equivalent.✓ Correct
  3. C.Quadratic form (b) is positive definite.✓ Correct
  4. D.Quadratic form (c) is positive definite.

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾
Let A be a real diagonal matrix with characteristic polynomial λ3−2λ2−λ+2\lambda^{3} - 2\lambda^{2} - \lambda + 2. Define a bilinear form ⟨v, w⟩ = vᵗAw on R3\mathbb{R}^{3}. Which of the following statements are true?
  1. A.A is positive definite.
  2. B.A2A^{2} is positive definite.✓ Correct
  3. C.There exists a nonzero v∈R3v \in \mathbb{R}^{3} such that ⟨v, v⟩ = 0.✓ Correct
  4. D.rank A = 2.

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾
Let Q(x,y,z)=2x2+2y2+2z2+2Q(x, y, z) = 2x^{2} + 2y^{2} + 2z^{2} + 2xy + 2yz + 2zx. Which of the following are true?
  1. A.The matrix of Q has eigenvalues 4, 1, 1.✓ Correct
  2. B.Q is positive definite.✓ Correct
  3. C.The signature of Q is (3, 0).✓ Correct
  4. D.Q(x, y, z) = 0 for some (x, y, z) ≠ 0.

The worked solution and the per-option analysis are in Trap Analysis.