MATH20F_Review

MATH20F_Review - MATH20F - REVIEW Solving Linear Equations...

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Unformatted text preview: MATH20F - REVIEW Solving Linear Equations To solve a system of linear equations, we put them into an augmented matrix and use row reductions to put them in reduced row echelon form: x 1- x 2 + 2 x 3- 2 x 4 = 2- x 1 + 2 x 2 + x 4 = 1 2 x 2 + 3 x 3 + x 4 = 5-→ 1- 1 2- 2 2- 1 2 1 1 2 3 1 5 row ops-→ Reduced Row Echelon Form z }| { 1 9 1 1 5 1 1- 3 1 -→ x 1 + 9 x 4 = 1 x 2 + 5 x 4 = 1 x 3- 3 x 4 = 1 x 4 free-→ x 1 =- 9 x 4 + 1 x 2 =- 5 x 4 + 1 x 3 = 3 x 4 + 1 x 4 = x 4 +-→ x 1 x 2 x 3 x 4 = x 4 - 9- 5 3 1 + 1 1 1 | {z } Parametric Vector Form Remember that reduced row echelon form requires every pivot column to be all zeros except one 1. Def. A set of vectors { v 1 , v 2 ,..., v p } is linearly dependent if the equation x 1 v 1 + x 2 v 2 +···+ x p v p = has a nontrivial solution (with not all the x i ’s being zero). If the set is not linearly dependent, it is linearly independent . The equation above is equivalent to the equation A x = 0, where A is the matrix with with v 1 ,..., v p as columns, so linear independence is equivalent to A having all pivot columns. Linear Transformations Def. A transformation is a function that sends vectors to vectors. In more formal terms, a transfor- mation T from R n to R m (written T : R n → R m ) is a function that is applied to a vector x ∈ R n as input and returns a vector T ( x ) ∈ R m as output. Def. A transformation T is linear if (i) T ( u + v ) = T ( u ) + T ( v ) for all u and v , (ii) T ( c u ) = cT ( u ) for all u and scalars c . Thm. If T : R n → R m is a linear transformation, there is a unique n × m matrix A such that T ( x ) = A x for all x ∈ R n . This matrix can by found by computing A = [ T ( e 1 ) ··· T ( e n )], where { e 1 ..., e n } is the standard basis for R n . Def. The standard matrix for the linear transformation T is the matrix from the above theorem. Def. A transformation T : R n → R m is onto if for each b ∈ R m , T ( x ) = b for at least one x ∈ R n . Def. A transformation T : R n → R m is one-to-one if for each b ∈ R m , T ( x ) = b for at most one x ∈ R n . Thm. Let T : R n → R m be a linear transformation and let A be the standard matrix for T . (i) T is onto if and only if the columns of A span R m (i.e. Col A = R m ). (ii) T is one-to-one if and only if the columns of A are linearly independent. 1 Inverses Def. An n × n matrix A is invertible if there is some n × n matrix C with the property that C A = I and AC = I . If C exists, then we call it the inverse and we write it as A- 1 . So by definition, A- 1 A = I and AA- 1 = I ....
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This note was uploaded on 05/24/2011 for the course MATH 20F taught by Professor Buss during the Spring '03 term at UCSD.

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MATH20F_Review - MATH20F - REVIEW Solving Linear Equations...

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