KEY THEOREMS KEY IDEASKEY ALGORITHMS LINKED TO EXAMPLES next

key theorems key ideaskey algorithms  n vectorsn vectors in an n dimensional vector space VECTOR SPACE independent span Solve system equations basisFind dot product coordinates Take matrix times vector  dimension dimension domain,null space, range of a linear mapping LINEAR MAPPINGWrite matrix equation domain null space range Find matrix for lin map Take product of matrices  detA  0 detA  0 matrix forFind inverse of matrix composition inverse Find determinant of matrix  similarity, similarity, eigenstuff EIGENSTUFFFind eigenstuffeigenstuff similaritySimilar diagonal matrixdiagonal

V is a vector space of dimension n. S = { v 1, v 2, v 3,..., v n } then S is INDEPENDENT if and only if S SPANS V. Return to outline

If T is a LINEAR MAPPING then: the dimension of the DOMAIN of T = the dimension of the NULL SPACE of T + the dimension of the RANGE of T Return to outline 0

A and B are SIMILAR matrices if and only if there exists a matrix P such that: B= P –1 A P If A is the matrix for T relative to the standard basis then B is the matrix for T relative to the columns of P If B is diagonal then the diagonal entries of B are eigenvalues and the columns of P are eigenvectors Return to outline

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Reduces to:

next Reduces to:

next

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( 3 ) 4 2 ( 2 ) = next

( 3 ) 4 2 ( 2 ) = = 9 return to outline

( 3 2 )( 1 ) = () next

( 3 2 )( 1 ) = ( 5 ) nextdot product of row 1 of matrix with vector = entry 1 of answer

( 3 2 )( 1 ) = ( 5 ) dot product of row 2 of matrix with vector = entry 2 of answer return to outline

System of linear equations: Equivalent matrix equation: return to outline

A toy maker manufactures bears and dolls. It takes 4 hours and costs $3 to make 1 bear. It takes 2 hours and costs $5 to make 1 doll. Find the matrix for T next

A toy maker manufactures bears and dolls. It takes 4 hours and costs $3 to make 1 bear. It takes 2 hours and costs $5 to make 1 doll. Find the matrix for T return to outline

( 3 2 )( 1 2 ) = () next

AB ( 3 2 )( 1 2 ) = ( 5 ) dot product of row 1 of A with column 1 of B = entry in row 1 column 1 of AB next

AB ( 3 2 )( 1 2 ) = ( 57 ) dot product of row 1 of A with column 2 of B = entry in row 1 column 2 of AB next

AB ( 3 2 )( 1 2 ) = ( 57 ) dot product of row 2 of A with column 1 of B = entry in row 2 column 1 of AB next

AB ( 3 2 )( 1 2 ) = ( 57 ) dot product of row 2 of A with column 2 of B = entry in row 2 column 2 of AB return to outline

Reduces to next

Reduces to A A -1 return to outline

next To find eigenvalues for A, solve for :

next To find eigenvalues for A, solve for : The eigenvalues are 2 and 4

next To find eigenvalues for A, solve for : The eigenvalues are 2 and 4 An eigenvector belonging to 2 is in the null space of 2I - A

next To find eigenvalues for A, solve for : The eigenvalues are 2 and 4 An eigenvector belonging to 2 is in the null space of 2I - A 2I - A

next To find eigenvalues for A, solve for : The eigenvalues are 2 and 4 An eigenvector belonging to 2 is in the null space of 2I - A 2I - A an eigenvector belonging to 2 is any nonzero multiple of

next To find eigenvalues for A, solve for : The eigenvalues are 2 and 4 eigenvectors are:

next The eigenvalues are 2 and 4 eigenvectors are: A is similar to the diagonal matrix B

The eigenvalues are 2 and 4 eigenvectors are: B = P –1 A P = return to outline

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A is an n  n matrix detA  0 iff A is nonsingular (invertible) iff The columns of A are a basis for R n iff The null space of A contains only the zero vector iff A is the matrix for a 1-1 linear transformation