Linear Algebra problem Topic Matrix Notation and Algebra Tr

Linear Algebra problem ... Topic: Matrix Notation and Algebra

True or False? a. If 3 vectors lie in the same plane, then there is a dependence relation between them. b. If a set of vectors is linearly dependent, then a vector in the set is a scalar multiple of one of the others. c. If a set of vectors is linearly dependent, then each vector in the set can be written as a linear combination of the others. d. If a set of vectors is linearly dependent, then there exists a vector in the set that can be written as a linear combination of the others. e. The columns of any 4 times 5 matrix are linearly dependent. f. A set of fewer than n vectors in R^n is linearly independent.

Solution

a) true

b) True

c) true

Let u, v and w be vectors, then

au +bv + cw = 0

since u,v,w are linearly dependent, so a,b and c are non zero, so

au= -bv - cw

u = (-b/a) v + ( -c/a)w

similary, we can show for v and w.

d) true

Let u, v and w be vectors, then

au +bv + cw = 0

since u,v,w are linearly dependent, so a,b and c are non zero, so

au= -bv - cw

u = (-b/a) v + ( -c/a)w

e) true

If A is 4x5 matrix, then rank A< min (4,5) =4

means A is singular matrix, so columns of A are linearly dependent.

f) false

since set A contains less than n elements, so

rank A < n that means A is singular. Therefore set A contains linearly dependent column elements.