Question Bank (E.M._Unit-1) (3).pdf

Karnavati University, Gandhinagar First Semester 2024-2025, (Engineering Mathematics-I) Question Bank (Unit-1) 1. Let V denote the set of ordered triples (x, y, z) and define addition in V as in R3. For each of the following definitions of scalar multiplication, decide whether V is a vector space. (i) a(x, y, z) = (ax, y, az) (ii) a(x, y, z) = (ax, 0, az) (iii) a(x, y, z) = (0, 0, 0) (iv) a(x, y, z) = (2ax, 2ay, 2az) 2. Are the following sets vector spaces with the indicated operations? If not, why not? (i) The set V of nonnegative real numbers; ordinary addition and scalar multiplication. (ii) The set V of all polynomials of degree ≥ 3, together with 0 ; operations of P. (iii) The set of all polynomials of degree ≤ 3; operations of P. (iv) The set {1, x, x2 ,...}; operations of P. a b (v) The set V of all 2 × 2 matrices of the form ; operations of M22. 0 c (vi) The set V of 2 × 2 matrices with equal column sums; operations of M22. (vii) The set V of 2 × 2 matrices with zero determinant; usual matrix operations. (viii) The set V of real numbers; usual operations. (ix) The set V of all ordered pairs (x, y) with the addition of R2 , but using scalar multiplication a(x, y) = (ax, −ay). (x) The set V of all ordered pairs (x, y) with the addition of R2 , but using scalar multiplication a(x, y) = (x, y) for all a in R. (xi) The set V of all 2 × 2 matrices whose entries sum to 0; operations of M22. (xii) The set V of all 2 × 2 matrices with the addition of M22 but scalar multiplication * defined by a ∗ X = aX T. 3. Let V be the set of all positive real numbers with vector addition being ordinary multi- plication, and scalar multiplication being a v = v a. Show that V is a vector space. 4. If V is the set of ordered pairs (x, y) of real numbers, show that it is a vector space with addition (x, y) ⊕ (x1 , y1 ) = (x + x1 , y + y1 + 1) and scalar multiplication a (x, y) = (ax, ay + a − 1). What is the zero vector in V. 5. Which of the following are subspaces of P3 ? Support your answer. (i) U = {f (x) | f (x) ∈ P3 , f (2) = 1} (ii) U = {xg(x) | g(x) ∈ P2 } (iii) U = {xg(x) | g(x) ∈ P3 } (iv) U = {xg(x) + (1 − x)h(x) | g(x) and h(x) ∈ P2 } (v) U = The set of all polynomials in P3 with constant term 0 (vi) U = {f (x) | f (x) ∈ P3 , deg f (x) = 3} 6. Which of the following are subspaces of M22 ? Support your answer. a b (i) U = a, b, and c in R} 0 c a b (ii) U = a + b = c + d; a, b, c, d in R} c d (iii) U = A | A ∈ M22 , A = AT (iv) U = {A | A ∈ M22 , AB = 0} , B a fixed 2 × 2 matrix (v) U = {A | A ∈ M22 , A2 = A} (vi) U = {A | A ∈ M22 , A is not invertible } 7. Which of the following are subspaces of F[0, 1] ? Support your answer. (i) U = {f | f (0) = 0} (ii) U = {f | f (0) = 1} (iii) U = {f | f (0) = f (1)} (iv) U = {f | f (x) ≥ 0 for all x in [0, 1]} (v) U = {f | f (x) = f (y) for all x and y in [0, 1]} (vi) U = {f | f (x + y) = f (x) + f (y) for all x and y in [0, 1]} R1 o (vii) U = {f | f is integrable and 0 f (x)dx = 0 8. Write each of the following as a linear combination of x + 1, x2 + x, and x2 + 2. (i) x2 + 3x + 2 (ii) 2x2 − 3x + 1 (iii) x2 + 1 (iv) x 9. Determine whether v lies in span {u, w} in each case. (i) v = 3x2 − 2x − 1; u = x2 + 1, w = x + 2 (ii) v = x, u = x2 + 1, w = x + 2 Page 2 1 3 1 −1 2 1 (iii) v = ;u = ,w = −1 1 2 1 1 0 1 −4 1 −1 2 1 (iv) v = ;u = ,w = 5 3 2 1 1 0 10. (i) Show that R3 is spanned by {(1, 0, 1), (1, 1, 0), (0, 1, 1)}. (ii) Show that P2 is spanned by {1 + 2x2 , 3x, 1 + x}. (iii) Show that M22 is spanned by 1 0 1 0 0 1 1 1 , , , 0 0 0 1 1 0 0 1 11. Is it possible that {(1, 2, 0), (1, 1, 1)} can span the subspace U = {(a, b, 0) | a and b in R} ? 12. Show that each of the following sets of vectors is independent. (i) {1 + x, 1 − x, x + x2 } in P2 (ii) {x2 , x + 1, 1 − x − x2 } in P2 1 1 1 0 0 0 0 1 (iii) , , , in M22. 0 0 1 0 1 −1 0 1 1 1 0 1 1 0 1 1 (iv) , , , in M22. 1 0 1 1 1 1 0 1 13. Which of the following subsets of V are independent? (i) V = P2 ; {x2 + 1, x + 1, x} (ii) V = P2 ; {x2 − x + 3, 2x2 + x + 5, x2 + 5x + 1} 1 1 1 0 1 0 (iii) V = M22 ; , , 0 1 1 1 0 1 −1 0 1 −1 1 1 0 −1 (iv) V = M2n ; , , , 0 −1 −1 1 1 1 −1 0 1 1 1 (v) V = F[1, 2]; x , x2 , x3 1 1 (vi) V = F[0, 1]; x2 +x−6 , x2 −5x+6 , x21−9 14. Find all values of a such that the following are independent in R3. (i) {(1, −1, 0), (a, 1, 0), (0, 2, 3)} (ii) {(2, a, 1), (1, 0, 1), (0, 1, 3)} 15. Show that the following are bases of the space V indicated. (i) {(1, 1, 0), (1, 0, 1), (0, 1, 1)}; V = R3 Page 3 (ii) {(−1, 1, 1), (1, −1, 1), (1, 1, −1)}; V = R3 1 0 0 1 1 1 1 0 (iii) , , , ; 0 1 1 0 0 1 0 0 (iv) {1 + x, x + x2 , x2 + x3 , x3 } ; V = P3 16. Exercise 7.1.1 Show that each of the following functions is a linear transformation. (i) T : R2 → R2 ; T (x, y) = (x, −y) (reflection in the x ax is) (ii) T : R3 → R3 ; T (x, y, z) = (x, y, −z) (reflection in the x − y plane) (iii) T : Mnn → Mnn ; T (A) = AT + A (iv) T : Pn → R; T [p(x)] = p(0) (v) T : Pn → R; T (r0 + r1 x + · · · + rn xn ) = rn (vi) T : Rn → R; T (x) = x · z, z a fixed vector in Rn (vii) T : Pn → Pn ; T [p(x)] = p(x + 1) (viii) T : Rn → V ; T (r1 , · · · , rn ) = r1 e1 + · · · + rn en where {e1 ,... , en } is a fixed basis of V (ix) T : V → R; T (r1 e1 + · · · + rn en ) = r1 , where {e1 ,... , en } is a fixed basis of V 17. In each case, show that T is not a linear transformation. (i) T : Mnn → R; T (A) = det A (ii) T : Mnm → R; T (A) = rank A (iii) T : R → R; T (x) = x2 (iv) T : V → V ; T (v) = v + u where u 6= 0 is a fixed vector in V (T is called the translation by u) 18. In each case, assume that T is a linear transformation. (i) If T : V → R and T (v1 ) = 1, T (v2 ) = −1, find T (3v1 − 5v2 ). (ii) If T : V → R and T (v1 ) = 2, T (v2 ) = −3, find T (3v1 + 2v2 ). 2 2 1 1 1 0 −1 (iii) If T : R → R and T = ,T = , find T. 3 1 1 1 3 2 2 1 0 1 1 1 (iv) If T : R → R and T = ,T = , find T. −1 1 1 0 −7 (v) If T : P2 → P2 and T (x + 1) = x, T (x − 1) = 1, T (x2 ) = 0, find T (2 + 3x − x2 ). (vi) If T : P2 → R and T (x + 2) = 1, T (1) = 5, T (x2 + x) = 0, find T (2 − x + 3x2 ). Page 4

Question Bank (E.M._Unit-1) (3).pdf

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