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# Class Note for MATH 215 with Professor Dostert at UA 3

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Date Created: 02/06/15

THE UNIVERSITY a OF ARIZONA Math 215 Introduction to Linear Algebra Section 32 Matrix Algebra Paul Dostert August 28 2008 Ac Matrix Addition amp Scalar Multiplication Let A7 B and C39 be matrices of the same size and let 0 and d be scalars Then a b C 0 6 f g h A B B A Commutativity A B C39 A B C Associativity A O A Additive identity A A O Additive inverse c A B cA cB Distributivity c d A cA dA Distributivity C dA Cd A 1A A To prove any of these properties we use the fact that addition and scalar multiplication is done componentwise This is the same way we prove these for vectors Just like with vectors we have the concept of a linear combination where cl ClAl C2A2 CkAk 7 ck are the coefficients of the linear combination Ac Matrix Addition amp Scalar Multiplication i 0 2 i 1 1 i 2 1 ExLetA1 0B1 1andC391 1 a S a near com Ina Ion O 7 an I D 3 2 l39 b39 t39 fA B d Cquot S a Inear com Ina Ion O 7 an b I D i 54 l39 b39 t39 fA B d Cquot c What is the span of A7 B and C We also have the concept of linear independence for matrices Matrices A17 714k of the same size are linearly independent if the only solution to 01A1CkAkO is the trivial solution 01 linearly dependent ck 0 If not then we say the AZ are Ex Are the matrices A7 B and 0 given in the previous exercise linearly independent Ac Matrix Multiplication As mentioned previously matrix multiplication is not necessarily commutative in fact it is rarely commutative 1 0 5 2 39 2 1 1 1 Let A7 B and C39 be matrices whose sizes make the indicated operation possible and let k be a scalar Then a A BC AB C39 Associativity b A B C AB AC Left distributivity c A B C AC BC Right distributivity d k AB kA B A kB e ImAAAInifAiSmgtltn Ex ShowAB7 BAforA andB Multiplicitive identity The proof of most of the above properties can be done by writing the matrices in terms of row or column vectors 10 1 1 Ex Verify each property for k 2 A 0 2 B 1 1 and A Matrix Transpose Let A and B be matrices whose sizes make the indicated operation possible and let k be a scalar Then a ATT A b A BT AT BT c MDT k AT d ABT BTAT e A7 T ATV for all nonnegative integers 7quot 0 2 1 1 Thm a If A is a square matrix then A AT is symmetric b For any matrix A AAT and ATA are symmetric Ex Prove both properties in general and verify for A 1 2 1 3 Ala Matlab Most of the examples discussed in this section can be solved using the ideas from our previous Matlab problems As an example we will use Matlab to verify the properties of the transpose for two random matrices We let A and B be n gtlt n for the given n Declare the size n 3 Create the random matrices of size n gtlt n A randnn B randnn Property a A A Property b AB A B Property c with k2 2A 2A Property d AB B A Property e with r 10 AA1 0 A quot1 0

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