Linear_Algebra_DPP_15_of_Lec_18_Saakaar_2.0_Batch_for_IIT_JAM_2024_Mathematicssaakaar-2-0-batch-for-iit-jam-2024-mathematics-327842mathematics-781061linear-algebra-266253.pdf

1 SAAKAAR 2.0 BATCH Linear Algebra DPP-15 1. Wh...

1 SAAKAAR 2.0 BATCH Linear Algebra DPP-15 1. Which of the following is not a subspace of R3? 4. Let M be the space of all 4×3 matrices with entries (1) {(a,b,c) | a,b,c rational} in the finite field of three elements. Then the number (2) (0,0,0)} of matrices of rank three in M is (3) {(a,a+b, –a+2b) | a,b real} (1) (34 – 3) (34 – 32) (34 – 33) (4) {(a,a – b,b) | a,b real} (2) (34 – 1) (34 – 2) (34 – 3) (3) (34 – 1) (34 – 3) (34 – 32) 2. Let V denote the vector space C5 [a,b] over R and (4) 34(34 – 1) (34 – 2)  d 4f d 2f  W  f  V : 4  2 2  f  0 .Then 5. The dimension of the vector space  dt dt  V={A=[aij](n×n) | aij∈C, aij= –aji} (1) dim(V) = ∞ and dim(W) = ∞ over field R is (2) dim(V) = ∞ and dim(W) = 4 (1) n2 (2) n2 – 1 (3) dim(V) = 6 and dim(W) = 5 (3) n – n 2 (4) n2/2 (4) dim(V) = 5 and dim(W) = 4 6. Consider the subspace 3. Let F3 be the field of 3 elements and let F3 × F3 be W={[aij]: aij=0if i is even} of all 10×10real matrices. the vector space over F3. The number of distinct Then the dimension of W is linearly dependent sets of the from {u, v}, where u, (1) 25 (2) 50 v ∈ F3 × F3 / {(0,0)} and u ≠ v is (3) 75 (4) 100 (1) 4 (2) 6 (3) 8 (4) 5  2 Answer Key 1. (1) 4. (3) 2. (2) 5. (3) 3. (1) 6. (2)  3 Hint and Solutions 1. (1)  a11 a12 a13  Scalar multiplication fails in option (1) a a 22 a 23  A   21 2. (2)  a 31 a 32 a 33    Let V denote the vector space C5 [a, b] over R.  a 41 a 42 a 43  C5[a, b] = {f ∈ C [a, b] | 5th derivative exist and is cts} As rank (A) = 3 ∴ dim V = ∞ ∴ No. of L.I. columns = 3  d 4 t 2d 2 t  ∴ A = [c1, c2, c3] ci∈ F4 W  f  V : 4  2  f  0  No. of L. I. subset S ⊆ V, |S| = k  dt dt  = (pn – 1) (pn – p) (pn – p2)…..(pn – p(k-1)) Dim W = 4 Here p = 3, n = 4, k = 3 Note: Solution Space of a nth order Linear ∴ No. of L. I. subset homogeneous differential equation forms a vector = (34 – 1) (34 – 3) (34 – 32) space of dimension n. 5. (3) 3. (1) Dimension of vector space of skew symmetric The total number of subsets in 𝐹3 × 𝐹3 − {(0,0)} matrix of n × n of real entries having exactly Two elements is 8𝐶2 = 28, also n2  n the number of Linearly independent subsets of  (32 −1)(32 −3) 2 order 2 is = 24 2! As dimension (C(R)) = 2 Therefore the number of Linearly dependent Dimension of vector space subset of the form {𝑢, 𝑣} in 𝐹3 × 𝐹3 − {(0,0)} is v = {A = [aij](n×n) | aij∈C, aij = –aji} 28 − 24 = 4 n2  n Over field R= dim(C(R)) 4. (3) 2  Let N = A  a ij  43  a ij  3  F  n2  n 2 × 2 = n2 – n X  A  M : rank  A   3 6. (2)  W= a ij  ;a  0 if is even 1010 ij  Dimension of W=10×5 PW Web/App - https://smart.link/7wwosivoicgd4 Library- https://smart.link/sdfez8ejd80if