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13-review

# 13-review - REVIEW HOURLY I Math 21b O Knill DEFINITIONS...

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REVIEW HOURLY I Math 21b, O. Knill DEFINITIONS. Linear subspace ~ 0 X , ~x, ~ y X, λ R ~x + ~ y X , λ~x X . Matrix A is a n × m matrix, it has m columns and n rows, maps R m to R n . Square matrix n × n matrix, maps R n to R n . Vector n × 1 matrix = column vector, 1 × n matrix = row vector. Linear transformation T : R n R m , ~x 7→ A~x , T ( ~x + ~ y ) = T ( ~x ) + T ( ~ y ) , T ( λ~x ) = λT ( ~x ). Column vector of A are images of standard basis vectors ~ e 1 , . . . ,~ e n . Linear system of equations A~x = ~ b , n equations, m unknowns. Consistent system A~x = ~ b : for every ~ b there is at least one solution ~x . Vector form of linear equation x 1 ~v 1 + . . . + x n ~v n = ~ b , ~v i columns of A . Matrix form of linear equation ~w i · ~x = b i , ~w i rows of A . Augmented matrix of A~x = ~ b is the matrix [ A | b ] which has one column more as A . Coefficient matrix of A~x = ~ b is the matrix A . Matrix multiplication [ AB ] ij = k A ik B kj , dot product of i -th row of A with j ’th column of B . Gauss-Jordan elimination A rref( A ) in row reduced echelon form. Gauss-Jordan elimination steps: swapping rows, scaling rows, adding rows to other rows. Row reduced echelon form : every nonzero row has leading 1, columns with leading 1 are 0 away from leading 1, every row with leading 1 has every rows above with leading 1 to the left. Pivot column column with leading 1 in rref( A ). Redundant column column with no leading 1 in rref( A ). Rank of matrix A . Number of leading 1 in rref( A ). It is equal to dim(im( A )). Nullety of matrix A . Is defined as dim(ker( A )).
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