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Row Space, Column Space, and Nullspace

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Row space , Column space , and NullspaceLinear AlgebraMATH 2010 Terminology:LetAbe the 2x4 matrixA=[2 3 1 04 5 6 2]Therow vectors ofAare[2 3 1 0][4 5 6 2](the rows ofA) in< vectors ofAare[24],[35],[ 16],[02](the columns ofA) in<2 Definition:LetAbe amxnmatrix (recallmis the number of rows andnis the number of columns ),then Therow spaceofAis the subspace of<nspanned by the row vectors ofA Thecolumn spaceofAis the subspace of<mspanned by the Column vectors ofA. Theorem:If amxnmatrixAis row-equivalent to amxnmatrixB, then the row space ofAis equalto the row space ofB. (NOT true for the Column space ) Theorem:If a matrixAis row-equivalent to a matrixBin row-echelon form, then the nonzero rowvectors ofBform a basis for the row space ofA. Example - Finding a Basis for Row SpaceLetA= 1 1 4 1 20 1 2 1 10 0 0 1 21 1 0 0 22 1 6 0 1 Find a basis for the row space must reduceA: 1 1 4 1 20 1 2 1 10 0 0 1 21 1 0 0 22 1 6 0 1 1 1 4 1 20 1 2 1 10 0 0 1 20 2 4 1 00 1 2 2 3 1 1 4 1 20 1 2 1 10 0 0 1 20 0 0 1 20 0 0 1 2

{ The row space of A is the subspace of <n spanned by the row vectors of A { The column space of A is the subspace of <m spanned by the column vectors of A. † Theorem: If a mxn matrix A is row-equivalent to a mxn matrix B, then the row space of A is equal to the row space of B. (NOT true for the column space)

  Space, Columns, Subspaces, Column space, And nullspace, Nullspace

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