Column space

Column space

Lessons

The column space of a matrix AA is a subspace of Rn\Bbb{R}^n.

Suppose the matrix AA is:

A=[v1v2vn] A=[v_1\;v_2\; \cdots \;v_n ]

where v1,v2,,vnv_1,v_2,\cdots,v_n are the columns of AA. Then the column space of AA is the set of vectors in C(A)C(A) which forms a linear combination of the columns of AA.

To see if a vector b\vec{b} is in the column space of AA, we need to see if b\vec{b} is a linear combination of the columns of AA. In other words,
b=x1v1+x2v2++xnvn \vec{b} =x_1 v_1+x_2 v_2+\cdots+x_n v_n

where x1,x2,,xnx_1,x_2,\cdots,x_n are solutions to the linear equation.

To find a basis for the column space of a matrix A, we:
1) Row reduce the matrix to echelon form.
2) Circle the columns with pivots in the row-reduced matrix.
3) Go back to the original matrix and circle the columns with the same positions.
4) Use those columns to write out the basis for C(A)C(A).

Note that the vectors in the basis are linearly independent.
  • 1.
    Column Space Overview:
    a)
    definition of the column space
    C(A)=C(A)= column space
    • A set of vectors which span{v1,v2,,vnv_1,v_2,\cdots,v_n}
    bϵC(A),\vec{b} \;\epsilon \;C(A), b=x1v1+x2v2++xnvn\vec{b}= x_1 v_1+x_2 v_2+\cdots+x_n v_n

    b)
    A vector in the column space
    b=x1v1+x2v2++xnvn\vec{b} =x_1 v_1+x_2 v_2+\cdots+x_n v_n
    • Changing to Ax=bAx=b and solve

    c)
    Finding a basis for the column space
    • Row reduce the matrix to echelon form.
    • Locate the columns with pivots in the row-reduced matrix.
    • Go back to the original matrix and find the columns with the same position.
    • Use those columns to write out the basis


  • 2.
    Finding if a vector is in the column space
    Let column space, vector A and vector b, finding if the vector is in the column space of A. Determine whether bb is in the column space of AA.

  • 3.
    Let column space, vector A and vector b, finding if the vector is in the column space of A. Determine whether bb is in the column space of AA.

  • 4.
    Finding a Basis for the Column Space
    Here is the matrix AA, and an echelon form of AA. Find a basis for C(A)C(A) (column space of AA).
    matrix A, and its echelon form

  • 5.
    Find a basis for the column space of AA if:
    Find a basis for the column space of A