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  # 111midtermsummary -          ...
  
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  MATH111 Midterm Summary A system of equations is consistent ⇐⇒ The system has a solution.(Unique or Infinitely many) There are 3 types of elementary row operations. Multiplying the row by a non-zero constant Interchanging two rows One row plus or minus a scalar multiple of other row. When A B , by a specific elementary row operation, B = EA , where E is the matrix in which the same elementary row operation performs on the identity matrix I . If A B , A and B are said to be row-equivalent . Given a linear transformation T , it has a matrix representation A . Then Vol( T ( P )) = | det( A ) | Vol( P ). Given a linear transformation T : R n R m , the standard matrix A = [ T ( e 1 ) . . . T ( e n )]. Homogenous system A x = 0 . The solution set can be written into the form s v 1 + t v 2 + . . . . If the corresponding system A x = b is consistent, then the solution set can be written into the form v + s v 1 + t v 2 + . . . . Let T : R n R m be linear. Let A be the standard matrix of T .
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  Unformatted text preview:           T is one-to-one m A x = has only trivial solution m Every column of A has a pivot position m The columns of A are linearly independent                                If A is an n × n matrix ←--------------→                                T is onto m A x = b has a solution for all b m Every row of A has a pivot position m The columns of A span R m                                • Inverse: A-1 = 1 det( A ) (adj( A )) [ A | I ]-→ [ I | A-1 ] • Determinant: A is invertible ⇐⇒ det A 6 = 0 1...
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