Final answer:
a. The columns of a matrix A are linearly independent if the equation Ax = 0 has the trivial solution. False. If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S. True. The columns of any 4x5 matrix are linearly dependent. True. If x and y are linearly independent, and if {x,y,z} is linearly dependent, then z is in Span {x,y}.
Step-by-step explanation:
a. True. The columns of a matrix A are linearly independent if the equation Ax = 0 has only the trivial solution. This means that when the matrix A is multiplied by the vector x and results in the zero vector, the only possibility is when all the entries of x are zero.
b. False. If a set S is linearly dependent, it means that there exists at least one vector in S that can be written as a linear combination of the other vectors in S. However, it does not mean that every vector in S can be written as a linear combination of the others.
c. True. In any 4x5 matrix, there are more columns than rows, which means that there are more vectors than the dimension of the vector space they belong to. Therefore, the columns of any 4x5 matrix are guaranteed to be linearly dependent.
d. True. If x and y are linearly independent, it means that neither vector can be written as a linear combination of the other. However, if {x, y, z} is linearly dependent, it means that one of the vectors can be expressed as a linear combination of the others. In this case, it must be z that can be written as a linear combination of x and y. Therefore, z is in Span {x, y}.