Practice Questions on Inner product, Bilinear and Quadratic Form with Answers

**Questions on Inner product, Bilinear and Quadratic Form with Answers**

**Q1. Let u,v, w be vectors in an inner-product space V, satisfying ||u|| = ||v|| = ||w|| = 2 and <u,v> = 0, <u,w> = l, <v,w> = -1. Then which of the following are true?**

(a) ||w + v – u|| = 2√2

(b) {^{1}/_{2} u , ^{1}/_{2} v} forms an orthonormal basis of a two dimensional subspace of V

(c) w and 4u – w are orthogonal to each other.

(d.) u, v, w necessarily linearly are independent.

**Q2. Let V denote the vector space of all polynomials over R of degree less than or equal to n. Which of the following defines a norm on V?**

(a) ||p||^{2} = |P(1)|^{2} +…+|p(n+1)|^{2} , p∈V

(b) ||p|| = sup_{t∈[0,1]} |p(t)| , p∈V

(c) ||p|| = _{0}∫^{1} |P(t)| dt , p∈V

(4) ||p|| = sup_{t∈[0,1]} |p'(t)| , p∈V

**Q3. Consider R ^{3} with the standard inner product. Let W be the subspace of R^{3} spanned by (1,0,-1). Which of the following is a basis for the orthogonal complement of W?**

(a) {(1,0.1), (0,1,0)}

(b) {(1,2,1), (0,1,1)}

(c) {(2,1,2), (4,2,4))

(d) {(2,-1.2), (1, 3, 1), (-1,-1,-1)}

**Q4. Consider the quadratic forms q and p given by q{x, y, z, w) = x ^{2} + y^{2} + z^{2} + bw^{2} and p(x, y, z, w) = x^{2} + y^{2} + cwz . Which of the following statements are true?**

(a) p and q are equivalent over C if b and care non-zero complex numbers.

(b) p and q are equivalent over R if b and c are non-zero real numbers.

(c) p and q are equivalent over R if b and c are non-zero real numbers with b negative.

(d) p and q are NOT equivalent over R if c 0

**Q5. Let V be the inner product space consisting of linear polynomials, p:[0,1] → R (i.e., V consists of polynomials p of the form p(x) = ax + b, a,b ∈R ), with the inner product defined by (p, q) = _{0}∫^{1}p{x)q(x)dx for p,q ∈V. An orthonormal basis of V is**

(a) {1, x}

(b) {1, x√3}

(c) {1, (2x-1)√3}

(d) {1, x-^{1}/_{2}}

**Q6. Consider the quadratic form q(x, y, z) = 4x ^{2} + y^{2} – z^{2} + 4xy – 2xz – yz over R. Which of the following statements about the range of vałues taken by q as x, y, z very over R, are true?**

(a) Range contains [1,∞)

(b) Range is contained in [0,∞)

(c) Range = R

(d) Range is contained in [-N, ∞) for some large natural number N depending on q

**Q7. Let J be the 3×3 matrix all of whose entries are 1. Then:**

(a) 0 and 3 are the only eigenvalues of J

(b) J is positive semi-definite, i.e., (Jx,x) ≥ 0 for all x∈R^{3}

(c) J is diagonalizable

(d) J is positive definite, i.e., (Jx,x) > 0 for all x∈R^{3} with x ≠ 0

**Q8. Let a _{ij} = a_{i}a_{j} , 1 ≤ i , j ≤ 1 where a_{1},…a_{n} are real numbers. Let A= [(a_{ij})] be the n×n matrix [(a_{ij})]. Then**

(a.) It is possible to choose a_{1},…,a_{n} so as to make the matrix A non singular

(b.) The matrix A is positive definite if (a_{1},…,a_{n}) Is a non zero vector

(c.) The matrix A is positive semi definite for all (a_{1},…,a_{n})

(d.) For all (a_{1},…,a_{n}) , zero is an eigenvalue of A.

**Q9. Suppose A, B are n×n positive definite matrices and I be the n×n identity matrix. Then which of the following are positive definite.**

(a.) A+B

(b.) ABA*

(c.) A^{2} + I

(d.) AB

**Q10. Let T be a linear transformation on the real vector space Rn over R such that T = λT for same λ∈R. Then**

(a) ||Tx|| = |λ| ||x|| for all x∈R^{n}

(b) If |Tx| = |x| for some non zero vector x∈Rn, then λ = ± 1

(c) T = |λI where I is the identity transformation on R^{n}

(d) If ||Tx|| > ||x|| for a non zero vector x∈R^{n}, then T is necessarily singular.

**Q11. The application of Gram- Schmidt process of orthonormalization to u _{1} =(1,1,0), u_{2} = (1,0,0), u_{3} =(1,1,1) yields**

(a) ^{1}/_{√2} (1,1,0), (1,0,0), (0,0,1)

(b) ^{1}/_{√2} (1,1,0), ^{1}/_{√2} (1,-1,0), ^{1}/_{√2} (1,1,1)

(c) (0,1,0), (1,0, 0), (0, 0,1)

(d) ^{1}/_{√2} (1,1,0), ^{1}/_{√2} (1,-1,0), ^{1}/_{√2} (0,0,1)

**Q12. Consider the basis {u _{1},u_{2},u_{3}} of R^{3} where u_{1} = (1,0,0). u_{2} = (1.1.0) , u_{3} = (1,1,1). Let {f_{1}•f_{2}•f_{3}} be the dual basis of (u1,u2,u3) and f be a linear functional defined by f(a,b,c) = a + b + c, (a,b,c)∈R^{3}. If f = a_{1}f_{1} + a_{2}f_{2} + a_{3}f_{3}. then (a_{1},a_{2},a_{3}) is**

(a) (1,2,3)

(b) (1,3,2)

(c) (2,3,1)

(d) (3,2,1)

**Q13. Consider R ^{3} with the standard inner product. Let S = {(1,1,1), (2.-1,2), (1,-2.1)} For a subset W of R3, let L(W) denote the linear span of W in R3. Then an orthonormal set T with L(S) = L(T) is**

(a) {^{1}/_{√3} (1,1,1) , ^{1}/_{√6} (1,-2,1)}

(b) {(1,0,0), (0,1,0), (0,0,1)}

(c) {^{1}/_{√3} (1,1,1) , ^{1}/_{√2} (1,-1,0)}

(d) {^{1}/_{√3} (1,1,1) , ^{1}/_{√2} (0,1,-1)}

**Q14. Let be a vector space (over R) of dimension 7 and Iet f: V → R be a non-zero linear functional. Let W be a linear subspace of V such that V = Ker(f) ⊕W where Ker(f) is the null space of f. What is the dimension of W?**

(a) 0

(b) 1

(c) 6

(d) None of these

**Q15. Consider the following statements:**

**P. Let A be a hermitian N × N positive definite matrix. Then, there exists a hermitian positive definite N × N matrix B such that B ^{2} = A.**

**Q. Let B be a nonsingular N × N matrix with real entries. Let B’ be its transpose. Then B’B is a symmetric and positive definite matrix.**

(a) P true Q false

(b) Q true P false

(c) Both are true

(d) Both are false

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*Answer Key*

*Answer Key*

01. | (a), (b), (c), (d) | 06. | (a), (c) | 11. | (d) |

02. | (a), (b), (c) | 07. | (a), (b), (c) | 12. | (a) |

03. | (a) | 08. | (c), (d) | 13. | (a) |

04. | (a), (c), (d) | 09. | (a), (b), (c) | 14. | (b) |

05. | (c) | 10. | (a), (b) | 15. | (c) |