In Exercises 21 and 22 mark each statement True or False Jus

In Exercises 21 and 22, mark each statement True or False. Justify each answer on the basis of a careful reading of the text. 21. a. The columns of a matrix A are linearly independent if the equation Ax = 0 has the trivial solution. b. If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S. c. The columns of any 4 times 5 matrix are linearly dependent. d. If x and y are linearly independent, and if {x, y, z} is linearly dependent, then z is in Span {x, y}. 22. a. Two vectors are linearly dependent if and only if they lie on a line through the origin. b. If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent. c. If x and y are linearly independent, and if z is in Span {x, y}, then {x, y, z} is linearly dependent. d. If a set in R^n is linearly dependent, then the set contains more vectors than there are entries in each vector. In Exercises 23-26, describe the possible echelon forms of the matrix. Use the notation of Example 1 in Section 1.2. 23. A is a 3 times 3 matrix with linearly independent columns.

Solution

22a. The statement is False. Two vectors are linearly dependent if one vector is a scalar multiple of the other vector. For this, the two vectors do not necessarily have to lie on a line through the origin.

b. False. For example, in R³ , the set { (0,1,1)^T, ( 0,2,2)^T} which contains fewer than 3 enties, is linearly dependent.

c. True. If z is in span {x, y}, then z is a linear combination of x and y so that the set { x, y, z} is linearly dependent.

d.False. A set containing fewer than n entries ( i.e. number of entries in each vector can also be linearly dependent as has been shown in b above.