Hi,
Just started week 1. Is there a difference between non-singular and complete? non-singular seems to suggest there is always 1 solution (so far in the videos), and the answers in videos/quizzes are coupled with complete, but I’m not sure what complete is; 2. what would be something that is complete but singular?
Hi @YodaKenobi!
A non-singular matrix is one that has a unique solution (invertible with a non-zero determinant). On the other hand, a complete set of vectors means that the vectors span the entire space, covering all possible solutions in that space.
A set of vectors can be complete but singular if the vectors span the space but are not linearly independent (while they cover the entire space, some vectors can be expressed as linear combinations of others).
Hope this helps! Feel free to ask if you need further assistance!
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The terminology of “complete”, “redundant” and “contradictory” was introduced by Prof Serrano in the lecture Systems of Sentences. Here’s a slide from that lecture:
I actually was a math grad student, although it was quite a long time ago, and I don’t ever remember that terminology being used in linear algebra. I think Prof Serrano has basically constructed that nomenclature for his explanations based on the idea of “sentences of information”, which is not strictly mathematical, but he uses it to help us understand systems of linear equations.
Singular and non-singular are standard mathematical definitions, which are defined in terms of the value of the determinant of the matrix describing the coefficients of the linear system. If the determinant is 0, then the system is “singular”. If it is non-zero, then the system is “non-singular”.
A non-singular system will have exactly one solution, so it is “complete” using Prof Serrano’s terminology.
A singular system can have either no solutions or an infinite number of solutions, so you need to do more work to figure out where it fits in the other namespace.
If this is still not seeming like a complete explanation to you, it might be a good idea to go back and watch the two lectures starting at the one I linked above.
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I’m not sure how the term “complete” applies to a set of vectors. My interpretation of what the Prof said is that complete is equivalent to non-singular for matrices. If you have a 3 x 3 matrix and at least one of the rows of the matrix is linearly dependent on the other rows, then the rank of the matrix is less than 3 and it is singular. In that case, the vectors do not form a basis for 3D space. But maybe I’m just missing your point and you mean that the set of vectors contains more than 3 elements. Sorry, it’s been > 6 months since I listened to all the lectures here: do you have a reference to where Prof Serrano defines what he means by “complete” for a set of vectors? I only found the place in the lecture I referred to above where he applies the term to sentences.
Hi @paulinpaloalto,
In this context, I interpreted complete to mean that the set of vectors spans the space, covering every direction within that space, regardless of whether the vectors are linearly independent.
For a matrix, when we say it’s non-singular, it implies it has full rank, meaning it spans the space with linearly independent vectors, forming a basis. However, a set of vectors can be complete but not independent, in which case it spans the space but has redundancy.
I’ll check the course videos for where Prof. Serrano explains complete in this context and let you know if I find anything.
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Hi, thanks for the replies @Alireza_Saei and @paulinpaloalto
- If I understand correctly singularity is defined based on the determinant of the matrix (just got to that part in the videos)
- linear
independencevsdependenceis defined based on whether a row is a linear combination of the other row(s) - Regarding the slide below: is it always the case a
dependentmatrix issingularand anindependentmatrix isnon-singular?
Yes, if the rows are all linearly independent, then the determinant will be non-zero and the matrix is non-singular. If they are dependent, then the determinant will be zero and the matrix will be singular. Later in this week, he introduces the concept of the rank of a matrix, which gives a more concrete way to express the independence or dependence.
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