Echelon form by P. Dewilde

Linear Algebra 3: Solving Ax = 0, Free Variables, and Pivot Variables, and Echelon Form

Last time we explored the null space, but how do we find the null space vectors? Additionally, how do we do these on nonsquare matrices?

Review of Null Space

Pivot Columns and Variables, Free Columns and Variables

The second and third columns are just multiples of the first.
We have two zeros in our pivot places!


Graphical Intuition for Ax =0

Solving Rectangular Systems for 0

The pivots are highlighted in green.
by P. Dewilde

15 y old learning about machine learning, as well as a lifelong naturalist. Climate activist in Vancouver. Writer. Visit me @

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