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I know that you can find the basis of ker by solving Ax = 0. If you can find the basis of the Image by taking the pivot columns can you do the same with ker with non-pivot columns?

WindBreeze
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    No, but yes. Paramaterize the solution. The resulting vectors that span are the basis vectors. You’re only really using the non-pivot columns to tell what the dimension of the kernel is. If you’re well practiced, you don’t have to perform this procedure and can just tell what the basis vectors will be. – DaveNine Mar 29 '19 at 20:36
  • See https://math.stackexchange.com/a/1521354/265466. If $A$ is not square, the column vectors have the wrong dimension to be in the kernel, but there’s a mechanical process for generating a kernel basis from the non-pivot columns. – amd Mar 29 '19 at 21:05

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