Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am stuck at this matrix: \[ \begin{pmatrix} 1 & 1 & 0 & 1 \\ -1 & 2 & 1 & 1 \\ -1 & 8 & 3 & 5 \end{pmatrix}. \] When going to normal form, we find this: \[ \begin{pmatrix} 1 & 1 & 0 & 1 \\ 0 & 3 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{pmatrix}. \]

Now it's obvious that the row space has the following basis: $\langle (1, 1, 0, 1), (0, 3, 1, 2) \rangle$

But how to determine the column space basis? I know by a theorem it must have dimension $2$.

share|improve this question
    
See here for an example of the pmatrix environment. –  Dylan Moreland Jan 21 '12 at 18:45

2 Answers 2

up vote 3 down vote accepted

Here's one way: Identify the columns of your fully reduced form that contain a non-zero leading row entry (the maroon entries in the reduced form below). For your reduced form, shown below, the leading non-zero row entries are colored maroon: $$ \left[ \matrix{ \color{maroon}1 & 1& 0 & 1\cr 0& \color{maroon}3& 1 & 2\cr 0& 0 &0 & 0}\right] $$ The corresponding columns are columns one and two.

A basis for the column space of the original matrix is given by the corresponding columns of the original matrix. So, here, the first two columns of the original matrix form a basis for its column space.

What's being used here is the following fact: elementary row operations on a matrix do not change the independence relations of the columns of the resulting matrix. Row operations can change the actual column space. So, once you have an echelon form, you can easily identify its independent columns (as above). Then the "same" columns of the original matrix will form a basis for its column space.

share|improve this answer
    
Thanks for your answer :) –  Kevin Jan 21 '12 at 19:07

Find the row reduced echelon form of Transpose[A] Transpose the result

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.