IIT JAM Mathematics Test Series 2023 : Linear Algebra – Inner product, Bilinear and Quadratic Form

Practice IIT JAM Mathematics Test Series for Free only On www.examflame.com and take your preparation to the another level. In the Series of Tests for IIT JAM 2023 this is the test of topic Inner product, Bilinear and Quadratic Form of the Chapter Linear Algebra. These Questions are prepared as per the Latest Syllabus of IIT JAM Mathematics 2023 . Practicing mock tests/Test series, you will get an idea about how and which type of Question will ask in the Examination. It also boost your confidence level. Also Solve IIT JAM Mathematics Previous Year Question Paper. And Don’t Forget to Shore with Your Friends.

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IIT JAM Mathematics Test Series 2023 : Linear Algebra – Inner product, Bilinear and Quadratic Form


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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