The following is a list of theorems and procedures from sections 5.1 and 5.2 that we have justified (fully or partially) during our classes. These are the results you may refer to during the exam. (Similar handout coming your way after we cover 5.3 and 5.4.)

1. Let be the matrix we get by putting in columns all vectors from the set of vectors in . Then is linearly independent if and only if the RREF of A has exactly many non-zero rows. Moreover, if the set is linearly dependent, than the vectors in are dependent in the same way as the corresponding columns in the RREF of A (for details see the lectures or the text.)

2. A set of vectors in is linearly independent if and only if the only solutions of is the trivial solution .

3. A set of many vectors in (note the two -s) is linearly independent if and only if the matrix we get by putting these vectors in columns is invertible.

4. If a subset of does not contain the zero vector, then it is not a subspace of . (But the converse is not true.)

5. The set of all solutions of a homogeneous system of equations is a subspace of .

6. The set spanned by a set of vectors in is a subspace of .

7. If some () is a linearly dependent on the other vectors in , then the sets and span the same subspace.

S.K.