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- Row space is the span of row vectors and a subset of
- Column space is span of column vectors and subset of
Theorem 3.6.1: Two row equivalent matrices have the same row space.
Theorem 3.6.2: The system has a solution iff is contained in the column space of .
Theorem 3.6.3: Let be a matrix.
- The linear system is consistent for every iff the column vectors of span .
- The system has at most one solution for every iff the column vectors of are linearly independent.
Properties of Row and Column Space
If Columns of are linearly independent, then
If columns of form a basis for , then
Corollary: the following are equivalent for a nonsingular square matrix
- columns form a basis for
- rows form a basis for
Given , where is in row echelon form; the column space of is NOT equal to the column space of .
- columns containing the lead variables form basis for the column space of
- above corresponds to columns of that form basis for 's column space
If is a matrix, then
can be converted to row-echelon form matrix , so .
Thus we have and equal the number of free variables: .
Thus (1, 2, 0, 3) and (0, 0, 1, 2) form a basis for the row space of
, and there are free variables.
The vectors (-2, 1, 0, 0) and (-3, 0, -2, 1) form a basis for the null space of , thus
If is a matrix, the dimension of the row space of equals the dimension of the column space of .
Let be the matrix obtained from by deleting columns corresponding to free vars, and let be obtained by deleting the same columns from . Both matrices are of size
, so if , then and because columns of are linearly independent. Therefore the columns of are linearly independent.
has columns, so the dimension of the column space of ≥ (and is also the dimension of the row space of
the column space of has the same dimension as the row space of ≥ the row space of has the same dimension as the column space of . By antisymmetry, the dimensions of the column and row spaces must be equal.
Thus form a basis for the column space of , and form a basis for the column space of .
form a basis for the row spaces of and (since they are equivalent).
The nullity of is thus the number of columns − num. lead variables = 5 − 3 = 2.
Let be vector spaces.
is a linear transformation if for all and for all ,
- (combination of 1 and 2)
Therefore, if , then , where is the image of .
Let be a linear operator on .
- (projection onto -axis)
- (reflect vector about -axis)
- (rotate by 90° CCW)