1.7 True/False

If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S

FALSE A set of vectors is said to be linearly dependent, if an only if there is at least one vector in the set such that it can be written as a linear combination of the others. (1,1), (2,2) and (5,4) are linear dependent, but we cannot express (5,4) as a combination of the other two

The columns of matrix A are linearly independent if the equation Ax=0 has the trivial solution

FALSE Matrix A is linearly independent if Ax=0 has ONLY the trivial solution Recall: Ax=0 always has the trivial solution (unique or infinite)

If a set contains fewer vectors than are entries in the vectors, then the set is linearly independent

FALSE The set can be linearly dependent

If a set in Rn is linearly independent, then the set contains more vectors than there are entries in each vector

FALSE The number of vectors in this set cannot exceed the number of entries in each vector

If x and y are linearly independent, and if {x,y,z} is linearly dependent, then z is in span {x,y}

TRUE If a set is linearly dependent, there exists at least one vector in the set such that it can be written as a linear combination of the others

The columns of any 4x5 matrix are linearly dependent

TRUE The columns of an nxp matrix are linearly dependent, if the number of variables is greater than the number of equations, p>n

Two vectors are linearly dependent if and only if they lie on a line through the origin

TRUE Two vectors are linearly dependent if and only if they lie on a line through the origin