Extra final review stuff

Extra final review stuff - V Consider the following system...

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The matrix equation: Ax = b Example: Find the solutions of Answer: (1) Solve the matrix equation ( Ax = b ) (2) Row reduce the augmented matrix (3) Solve the vector equation (4) This is equivalent to asking if is in the Span How to multiply matrices: How to find inverse of 3 x 3 matrixes 1. Find inverse of matrix 2. 3. 4. Definition of Vector Space: a nonempty set of vectors that must satisfy these ten axioms. The axioms must hold for all vectors and all constants 1. The sum of u and v , denoted by u + v , is in . 2. u + v = v + u 3. ( u + v ) + w = u + ( v + w ). 4. There is a zero vector 0 in such that u + 0 = u . 5. For each u in , there is a vector –u in such that u + ( -u ) = 0 . 6. The scalar multiple of u by c , denoted by c u , is in . 7. c ( u + v ) = c ( u ) + c ( v ) 8. ( c + d ) u = c u + d u . 9. c ( d u ) = ( cd ) u . 10. 1 u = u . Theorem: if are in a vector space V , then Span is a subspace of
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Unformatted text preview: V . Consider the following system of homogeneous equations: In matrix form this system is written as A x = , where We call the set of x that satisfy A x = the null space of the matrix A . Definition of Null Space: the set of all solutions to the homogenous equation A x = . In set notation: Theorem: The null space of an matrix A is a subspace of . Equivalently, the set of all solutions to a system of m homogeneous linear equations in n unknowns is a subspace of . Definition of Column Space: of an matrix A , written as Col A , is the set of all linear combinations of the columns of A . If , then Theorem: The column space of an matrix A is a subspace of . Span is linear if Nul A only contains the vector...
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Extra final review stuff - V Consider the following system...

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