How to Express Null(A) in Cartesian Form?

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SUMMARY

The discussion focuses on expressing the null space of matrix A in Cartesian form. The user derives the null space as X = (-2; 1; 1)*x3, indicating that Null(A) geometrically represents a line through the origin and the vector (-2, 1, 1). The user seeks clarification on whether this representation suffices or if it should also be expressed as the line -x = 2y = 2z for completeness. The matrix A is defined as A = (1 0 2; 0 3 -3; 4 2 6), and the null space is confirmed through row reduction and back substitution.

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  • Understanding of linear algebra concepts, specifically null spaces.
  • Familiarity with matrix operations, including row reduction.
  • Knowledge of Cartesian coordinates and vector representation.
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  • Study the process of finding the null space of matrices using row reduction techniques.
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators seeking to clarify concepts related to null spaces and their geometric interpretations.

canyonist
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Hi there,

just a pretty straight forward query I need cleared up...

If a question asks for the null space of A in cartesian form how do I set it out?

This is what I've got:

X = ( -2; 1; 1)*x3 for all values of x3

Therefore, Null(A) corresponds geometrically to the line through the origin and (-2, 1, 1).

Now do I need to say that this is equivelant to -x = 2y = 2z for it to be in cartesian form or will the above suffice?

thanks for the help
 
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I'm a bit confused. What is A, and how does that compare to X? Is X supposed to be the one dimensional vector space through the vector (-2,1,1)?
 
Yes, so AX = 0


where X = ( x1; x2; x3) and A = (1 0 2; 0 3 -3; 4 2 6)

which can be expressed as ( -2; 1; 1)*x3 after row reduction and back substitution.

I need to know whether I can leave it at that, or express it as the line: -x = 2y = 2z
 

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