1. 2 2.9 © 2016 Pearson Education, Inc. Matrix Algebra DIMENSION AND RANK

2. Slide 2.9- 2 © 2016 Pearson Education, Inc. COORDINATE SYSTEMS

3. Slide 2.9- 3 © 2016 Pearson Education, Inc. COORDINATE SYSTEMS

4. Slide 2.9- 4 © 2016 Pearson Education, Inc. COORDINATE SYSTEMS

5. Slide 2.9- 5 © 2016 Pearson Education, Inc. COORDINATE SYSTEMS

6. Slide 2.9- 6 © 2016 Pearson Education, Inc. COORDINATE SYSTEMS

7. Slide 2.9- 7 © 2016 Pearson Education, Inc. THE DIMESION OF A SUBSPACE  Definition: The dimension of a nonzero subspace H, denoted by dim H, is the number of vectors in any basis for H. The dimension of the zero subspace {0} is defined to be zero.  Definition: The rank of a matrix A, denoted by rank A, is the dimension of the column space of A.

8. Slide 2.9- 8 © 2016 Pearson Education, Inc. THE DIMESION OF A SUBSPACE

9. Slide 2.9- 9 © 2016 Pearson Education, Inc. THE DIMESION OF A SUBSPACE

10. Slide 2.9- 10 © 2016 Pearson Education, Inc. RANK AND THE INVERTIBLE MATRIX THEOREM

11. Slide 2.9- 11 © 2016 Pearson Education, Inc. RANK AND THE INVERTIBLE MATRIX THEOREM

12. Slide 2.9- 12 © 2016 Pearson Education, Inc. RANK AND THE INVERTIBLE MATRIX THEOREM