MA1522 4.1 Column Space, Row Space, and Nullspace

Column and Row Space

given a mxn matrix A, each row is a N vector, and each column is a m vector.

The row space of A, is the subspace of Rⁿ spanned by the rows of A.

Row(A) = span{(a11 a12 ... a1n), (a21, a22,...a2n), ...(am1, am2..amn)}

*can write the vectors in the row space as column vectors

The column space of A, is the subspace of R^m spanned by the columns of A

Col(A) = span{ (a11; a21;...am1;), (a12; a22;...am2;),...(a1n; a2n;...amn;)}

Basis for row space

If a matrix R is in reduced row echelon form, then the nonzero rows of R form a basis for its row space.

*By preservation of Row(A) under EROs, Row(A) = Row(rref(A))

Elementary row operation's affect on Row space

Let A be the pre-ERO matrix and B be the post-ERO matrix

1) Ri ←→ Rj; Exchanging two rows: Row(A) = Row(B)

2) cRi; multiplying a row by some scalar; Row(A) = Row(B). (c≠0)

3) Ri + cRj; adding a multiple of some row to another: Row(A) = Row(B)

=> EROs have no effect on Row(A)/preserve Row(A)

=> if A and B are row equivalent matrices, Row(A) = Row(B)

Basis for Column Space

Suppose R is the rref of a matrix A. Then the columns of A corresponding to the pivot columns in R form a basis for the column space of A. (Take the original columns from A though)

Row operations & Linear relations

Row operations preserve linear relations between columns.

Let A = (a1 a2.... an) and B = (b1 b2 .... bn) be row equivalent mxn matrices, where ai and bi is the ith column of A and B, respectively, for i = 1,...,n. Then for any coefficients c1, c2,...cn

c1a1+c2a2+...+cnan = 0

iff

c1b1+c2b2+...+cnbn = 0

Effect of EROs on column space

1)Elementary row operations do not preserve the column space of a matrix. Col(A) may be changed when performing EROs so we have to take the columns from the original matrix for the basis.

2)Row operations do not preserve linear relations between rows. Cannot tell which rows are independent/dependent or how the rows are dependent on each other

Column space and consistency of a linear system

The column space is the set of vectors such that Ax = v is consistent, or the set of vectors v= Au for some u.

Col(A) = { v = Au | u∈R^k} = {v | Ax = v is consistent}

Nullspace & Nullity

The null space of a mxn matrix A is the solution space to the homogeneous system Ax = 0, with coefficient matrix A. It is denoted as

Null(A) = { v∈Rⁿ | Av = 0}

Nullity(A) = dim(Null(A)), the dimension of the nullspace of A.

Basis for Nullspace

rref the coefficient matrix A of Av=0

then, solve the linear system (assign parameters to nonpivot columns, obtain the general solution by expressing x1, x2...xn in terms of r, s, t etc), the vectors of the general solution form the basis.

Nullity and Invertibility

If A is a square matrix,

Nullity(A) = 0 iff A is invertible