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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 .

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MATRIX ALGEBRA

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  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,

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