194 Chapter 4 General Vector spaces Concept Review Trivial solution Linearly dep

Question

194 Chapter 4 General Vector spaces Concept Review Trivial solution Linearly dependent set Linearly independent set Wronskian Skills Determine whether a set of vectors is linearly Use the Wronskian to show that a set of functions is independent or linearly dependent. linearly independent. Express one vector in a linearly dependent set a linear combination of the other vectors in the set. True-False Exercises In parts (a) (h) determine whether the statement is true or (e) If vi.....vn are linearly dependent nonzero vectors, false, and justify your answer. then at least one vector vk is a unique linear combina- (a) A set containing a single vector is linearly independent tion of v vk-1 (b) The set of vectors (v kv) is linearly dependent for ev-L (f) The set of 2 x 2 matrices that contain exactly two l's and two 0's is a linearly independent set in Ma ery scalar k. three polynomials (x 1)(x 2), x(x 2), and x (g The (c) Every linearly dependent set contains the zero vector. x(x 1) are linearly independent. L (d) If the set of vectors (vi, v3) is linearly indepen- (h) The functions fi and fi are linearly dependent if there dent, then (kvi, kv2, kv3) is also linearly independent so that ki fi(x) k2 f20x) 0 for is a real number x for every nonzero scalar k. some scalars ki and k2. 4.4 Coordinates and Basis and the We usually think of a line as being one-dimensional, a plane two-dimensional, next to minnal, It is goal as the of this section and llss coordinate

Explanation / Answer

(a) The statement is false. The set containing the single vector 0 is linearly dependenyt as k*0 = 0 even if k 0.

(b) The statement is true as (-k)v +kv = 0.

(c) The statement is false. For example, the set V ={(1,1),(2,2)} is linearly dependent as -2(1,1)+(2,2) = (0,0). The set V does not contain the 0 vector.

(d) The statement is true. If the set {kv1,kv2,kv3} is linearly dependent, then there exist scalars a,b,c (not all 0) such that akv1 +bkv2+ckv3 = 0. Then k(av1 +bv2+cv3) = 0 so that av1 +bv2+cv3 = 0. This means that v1,v2,v3 are linearly dependent, a contradiction.

(e) The statement is false. If v1,v2,…,vn are linearly dependent vectors, then at least one vector vk is a linear combination of the remaining vectors i.e., v1,v2,…,vk-1,vk+1,…,vn and NOT v1,v2,…,vk-1 .

(f) The statement is false. The set of 2x2 matrices that has 2 1’s and 2 0’s is e1+e2, e1+e3, e1+e4, e2+e3, e2+e4, e3+e4, where e1,e2,e3,e4 denote the standard basis 2x2 matrices in M22. Then e2+e4 = -(e1+e3) + (e1+e4) + (e2+e3).

(g) The statement is true. (x-1)(x+2) = x2 -3x-2,x(x+2) = x2+2x, x(x-1) = x2-x. The matrix oif coefficients is A =

1

1

1

-3

2

1

-2

0

0

The RREF of A is I3. Hence the given vectors are linearly independent.

(h) The statement is false. If both of k1 and k2 are not 0, then the statement is true.

1

1

1

-3

2

1

-2

0

0

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