## Lecture notes

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## Lecture notes

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Lecture Notes

- Pages:
- 6
- School:
- Boston University
- Course:
- Ma 242 - Linear Algebra

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MA 242 September 28 2007 Two theorems on mapping properties of linear transformations Theorem Let T Rn Rm be a linear transformation Then T is one to one if and only if the only solution to T x 0 is the trivial solution Theorem Let T Rn Rm be a linear transformation and let A be its standard matrix representation Then 1 T maps Rn onto Rm if and only if the columns of A span Rm and 2 T is one to one if and only if the columns of A are linearly independent 1 MA 242 September 28 2007 Matrix algebra Matrices have an unusual algebraic structure associated to them Matrix addition and scalar multiplication should remind you of vector addition and scalar multiplication in Rn Definitions 1 Let A and B be two m n matrices Then the sum of A and B is the m n matrix C where cij aij bij 2 Let A be an m n matrix and r be a real number Then the scalar multiple rA is the matrix whose ij entry is raij In other words every entry of A is multiplied by r Theorem 1 Let A B and C be m n matrices and let r and s be real numbers Then 1 A B B A 2 A B C A B C 3 A 0 A 4 r A B rA rB 5 r s A rA sA 6 r sA rs A Matrix multiplication is more subtle than addition or scalar multiplication and there are two ways to define it One way is algebraic The other is geometric 2 MA 242 September 28 2007 Here is the geometric definition Let A Rn Rm and B Rp Rn be linear transformations What is the matrix representation of the composition A B Note that this composition has B as the first transformation 3 MA 242 September 28 2007 Definition The product C of A and B is the matrix C AB1 AB2 ABp Let s check the sizes of all of the matrices involved Example Let A R2 R2 be rotation by 45 and let B R2 R2 be the matrix transformation determined by the matrix 2 1 1 1 What is the matrix representation for the transformation A B 4 MA 242 September 28 2007 Row column dot product definition The columns of AB are linear combinations of the columns of A In fact consider the jth column of AB Row column rule AB ij ai1 b1j ai2

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