lec1_4

lec1_4 - vector space (see chapter 4). 1.2. Some...

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NOTES ON SECTION 1 . 4 MA265, SECTIONS 41, 52 1. Matrix algebra We can add and multiply matrices, so we can try to do algebra with matrices. The algebra of matrices shares many things in common with the algebra of numbers we are used to (which we expect because addition and scalar multiplication is done entry-wise), but some things are very different (which we also expect because matrix multiplication is defined in an odd way - notice that matrix multiplication is not even well defined! when the inner dimensions disagree). 1.1. Vector space properties. Theorem 1. The following algebraic identities hold (whenever the size of the matrices involved allow for one to perform the opertations required to compute either side of the identity): (The addition axioms = matrices form an “abelian group”.) A + 0 = A A + B = B + A ( A + B ) + C = A + ( B + C ) For each matrix A , there is another matrix D such that A + D = 0 . From now on, we dentoe this matrix D by - A instead. (The scalar multiplication axioms - r,s are scalars here.) 1 · A = A r ( sA ) = ( rs ) A ( r + s ) A = rA + sA r ( A + B ) = rA + rB These properties taken together say that the space of matrices forms a
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Unformatted text preview: vector space (see chapter 4). 1.2. Some multiplication properties. Some more properties involving multiplication: Theorem 2. ( AB ) C = A ( BC ) ( A + B ) C = AC + BC and A ( B + C ) = AB + AC 1.3. Things which fail. However, here are some things which are FALSE! : It is not true that AB = BA . For example, 1 0 0 0 0 1 0 0 6 = 0 1 0 0 1 0 0 0 AB = 0 doesnt mean that either A = 0 or B = 0, for example 0 1 0 0 1 0 0 0 = 0 0 0 0 AB = AC might be true even if B 6 = C and A 6 = 0. For instance 1 0 0 0 1 1 1 1 = 1 0 0 0 1 1 0 0 but 1 1 1 1 6 = 1 1 0 0 1 (so you cant just cancel the matrix) 1 0 0 0 1.4. Algbraic properties of the transpose. Theorem 3. ( A T ) T = A ( A + B ) T = A T + B T ( AB ) T = B T A T NOTE THIS ONE CAREFULLY!!! ( rA ) T = rA T 2...
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lec1_4 - vector space (see chapter 4). 1.2. Some...

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