Questions on Inner product, Bilinear and Quadratic Form with Answers

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

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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|| = supt∈[0,1] |p(t)| , p∈V
(c) ||p|| = 01 |P(t)| dt , p∈V
(4) ||p|| = supt∈[0,1] |p'(t)| , p∈V


Q3. Consider R3 with the standard inner product. Let W be the subspace of R3 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) = x2 + y2 + z2 + bw2 and p(x, y, z, w) = x2 + y2 + 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) = 01p{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) = 4x2 + y2 – z2 + 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∈R3
(c) J is diagonalizable
(d) J is positive definite, i.e., (Jx,x) > 0 for all x∈R3 with x ≠ 0


Q8. Let aij = aiaj , 1 ≤ i , j ≤ 1 where a1,…an are real numbers. Let A= [(aij)] be the n×n matrix [(aij)]. Then

(a.) It is possible to choose a1,…,an so as to make the matrix A non singular
(b.) The matrix A is positive definite if (a1,…,an) Is a non zero vector
(c.) The matrix A is positive semi definite for all (a1,…,an)
(d.) For all (a1,…,an) , 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.) A2 + 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∈Rn
(b) If |Tx| = |x| for some non zero vector x∈Rn, then λ = ± 1
(c) T = |λI where I is the identity transformation on Rn
(d) If ||Tx|| > ||x|| for a non zero vector x∈Rn, then T is necessarily singular.


Q11. The application of Gram- Schmidt process of orthonormalization to u1 =(1,1,0), u2 = (1,0,0), u3 =(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 {u1,u2,u3} of R3 where u1 = (1,0,0). u2 = (1.1.0) , u3 = (1,1,1). Let {f1•f2•f3} 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)∈R3. If f = a1f1 + a2f2 + a3f3. then (a1,a2,a3) is

(a) (1,2,3)
(b) (1,3,2)
(c) (2,3,1)
(d) (3,2,1)


Q13. Consider R3 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 B2 = 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

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)

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