1 / 11

MATRIX ALGEBRA

MATRIX ALGEBRA. A linear system with e quations presents us with these three objects: the coefficient matrix . the u nkown vector . the constant vector .

zahur
Télécharger la présentation

MATRIX ALGEBRA

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. MATRIX ALGEBRA A linear system with equations presents us with these three objects: • the coefficient matrix. • the unkownvector. • the constant vector . Actually all three of these objects are matrices, we just call very skinny matrices “vectors”, but vectors are matrices !

  2. So we are just going to study matrices for awhile, no matter how skinny You remember our four friends, the algebraic operations . Let’s see how many of them we can extend to matrices. to vectors, provided they are We take our clue from vectors and define with the following definition.

  3. Definition. . We define This definition is extremely easy to apply,it needs no further elaboration. The algebraic operation What about It’s a free country, we could define

  4. where and we would not be wrong, just … With the wisdom of hindsight we give the Definition. We define , where

  5. It is important at this time to pay close attention to the (note the common middle dimension, it says number of columns of first = number of rows of second) Pictorially we have the figure shown in the next slide

  6. (we assume Do a few examples by yourselves!

  7. Henceforth we will, for quite some time, restrict ourselves to the case So that the product is … Recall that we have called one-column matrices . The matrix equation is re-written as where both Writing out long-hand the equation we get the familiar linear system

  8. (remember?) (OK, then I wrote Looking carefully at the long-hand system we note that there are three equivalent ways to write it

  9. Long-hand. • Let Then the system can be written as • In concise form Comparing 2 and 3 we get the following beautiful theorem (Theorem 4, p. 37) which ties in the span of the column vectors of Here goes the theorem:

  10. Theorem. The following four statements are equivalent: • For every Here is how to structure the proof:

  11. Each equivalence can be proved independently of the others. The second (from left) and third equivalences are obvious, to row-reduced echelon form. Here is a first consequence theorem. Corollary. Let (The converse is false,

More Related