Review2 - Review sheet for exam 2 - Math 291, Fall, 2007...

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Review sheet for exam 2 - Math 291, Fall, 2007 Definitions : Subspace, linearly dependent/independent sets of vectors, basis, span, null space, row space, column space, dimension, rank, nullity, linear transformations, Ker( f ), Range( f ), eigen- values and eigenvectors, characteristic polynomial of the matrix A , diagonalizable matrix, coordi- nates of the vector v in some basis, change of basis matrix from { e 1 , . . . , e n } to { f 1 , . . . , f n } . Main results : Let A be m × n and f A : R n R m the corresponding linear transformation. Then Ker( f A ) is a subspace of R n and Range( f A ) is a subspace of R m . Moreover, x Ker( f A ) ⇐⇒ A x = 0 ⇐⇒ x is in the null space of A ⇐⇒ x is a solution to the homogeneous equation. y Range( f A ) ⇐⇒ the linear system of equations A x = y has a solution x ⇐⇒ y is a linear combination of the columns of A (namely A x ) ⇐⇒ y the column space of A . That is, Range(
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This note was uploaded on 05/23/2008 for the course MATH 290/291 taught by Professor Lerner during the Spring '08 term at Kansas.

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