Lecture 9 A basic consists of vectors in the space Vectors independent The vectors span the space Row Space Nullspace Column Space A rank 1 m 2 n 3 R Dimension of the row space is 1 The row space is a line in R of all multiples of 1 4 6 A rank 1 m 2 n 3 R Dimension of the nullspace is 2 The nullspace is a plane in R of all the linear combinations of the two special solutions to Rx 0 A rank 1 m 2 n 3 R Dimension of the column space is 1 The column space is the line in R of all the linear combination of the columns Nullspace of A Transpose A rank 1 m 2 n 3 R Dimension of the left nullspace is 2 The left null space is the line in R of all the linear combination of the one special solution to A x 0 If the rows of A are all independent Row space basis is one Left nullspace basis is empty If the matrix is square and invertible The column space is all of R The null space is the zero vector The row space is all of R The left null space is the zero vector Note Row rank Column rank
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