Decide whether each of the following sets of vectors is linearly independent. a.

Question

Decide whether each of the following sets of vectors is linearly independent. a. {(1, 4), (2, 9)} R^2 b. {(1, 4, 0), (2, 9, 0)} R^3 c. {(1, 4, 0), (2, 9, 0), (3, -2, 0)} R^3 d. {(1, 1, 1), (2, 3, 3), (0, 1, 2)} R^3 e. {(1, 1, 1, 3), (1, 1, 3, 1), (1, 3, 1, 1), (3, 1, 1, 1)} R^4 f. {(1, 1, 1, -3), (1, 1, -3, 1), (1, -3, 1, 1, ), (-3, 1, 1, 1)} R^4 Decide whether the following sets of vectors give a is for the indicated space. a. {(1, 2, 1), (2, 4, 5), (1, 2, 3)}: R^3 b. {(1, 0, 1) (1, 2, 4), (2.2, 5), (2, 2, -1)}: R^3 c. {(1, 0, 2, 3), (0, 1, 1, 1), (1, 1, 4, 4)}: R^4 d. {(1, 0, 2, 3, ), (0, 1, 1, 1), (1, 1, 4, 4, ), (2, -2, 1, 2)}: R^4 In each case, check that {v_1, ..., v_n} is a basis for R^n and give the coordinates of the given vector b elementof R^n with respect to that basis. a. v_1 = [2 3], v_2 = [3 5]; b = [3 4] b. v_1 = [1 0 3], v_2 = [1 2 2], v_3 = [1 3 2]; b = [1 1 2] c. v_1 = [1 0 1], v_2 = [1 1 2], v_3 = [1 1 1]; b = [3 0 1] d. v_1 = [1 0 0 0], v_2 = [1 1 0 0], v_3 = [1 1 1 1], v_4 = [1 1 3 4]; b = [2 0 1 1] Following Example 10. for each of the following matrices A, give a basis for each of the subspaces R(A), C(A), N(A), and N(A^T). a. A = [3 -1 6 -2 -9 3] b. A = [1 1 0 2 1 1 1 -1 2] c. A = [1 1 1 1 2 0 1 1 1 1 0 2] d. A = [1 2 -1 0 2 4 -1 -1] Give a basis for the orthogonal complement of the subspace W R^4 spanned by (1, 1, 1, 2) and (1, -1, 5, 2).

Explanation / Answer

Given v1= ( 1,0,1)T ,v2=( 1,1,2)T and v3= ( 1,1,1) T

to verify weter tese 3 vectors are Linearly independent let us evaluate te determinant formed by tese 3 vectors

| 1 1 1

0 1 1

1 2 1 | = 1( -1) -1( -1) +1(-1) =-1 not equal to 0 .

ence te vectors are LI

B= ( 3,0,1) = linear combination of te 3vectors

=a ( 1,0,1) + b( 1,1,2)+c( 1,1,1)

equating te respective coords

a+b+c= 3 ----(1)

b+c= 0----(2)

a+2b+c=1---(3) from eqns 1 and 3 we get b= - 2 , from 2 we get c=2 and from 1 we get a= 3

a=3 , b=-2 ,c=2

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