Why is the set of all Real Upper Triangular Square matrices not a vector space?

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My textbook indicates that the set of all upper triangular n ✕ n matrices is a real vector space, but the set of all upper triangular square matrices is not a real vector space.



Why is there a difference between the two? Shouldn't the upper triangular square matrix set also be a vector space?




linear-algebra matrices vector-spaces




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  • I did. I used the exact same phrasing of the question from the online course I'm taking. I attached pictures to demonstrate my point.
    – Shayan Hemmati
    Aug 6 at 21:16










  • did the pictures provide any clarity or still a typo?
    – Shayan Hemmati
    Aug 6 at 21:19










  • Perfect! thank you for clarification
    – Shayan Hemmati
    Aug 6 at 21:23










  • Pjonin gave a good explanation, of course for the second one if we are dealing with square matrices of different size then "the set of all upper triangular square matrices is not a real vector space" is true of course!
    – gimusi
    Aug 6 at 21:28














up vote
1
down vote

favorite












My textbook indicates that the set of all upper triangular n ✕ n matrices is a real vector space, but the set of all upper triangular square matrices is not a real vector space.



Why is there a difference between the two? Shouldn't the upper triangular square matrix set also be a vector space?




linear-algebra matrices vector-spaces




share|cite|improve this question





















  • I did. I used the exact same phrasing of the question from the online course I'm taking. I attached pictures to demonstrate my point.
    – Shayan Hemmati
    Aug 6 at 21:16










  • did the pictures provide any clarity or still a typo?
    – Shayan Hemmati
    Aug 6 at 21:19










  • Perfect! thank you for clarification
    – Shayan Hemmati
    Aug 6 at 21:23










  • Pjonin gave a good explanation, of course for the second one if we are dealing with square matrices of different size then "the set of all upper triangular square matrices is not a real vector space" is true of course!
    – gimusi
    Aug 6 at 21:28












up vote
1
down vote

favorite









up vote
1
down vote

favorite











My textbook indicates that the set of all upper triangular n ✕ n matrices is a real vector space, but the set of all upper triangular square matrices is not a real vector space.



Why is there a difference between the two? Shouldn't the upper triangular square matrix set also be a vector space?




linear-algebra matrices vector-spaces




share|cite|improve this question













My textbook indicates that the set of all upper triangular n ✕ n matrices is a real vector space, but the set of all upper triangular square matrices is not a real vector space.



Why is there a difference between the two? Shouldn't the upper triangular square matrix set also be a vector space?





linear-algebra matrices vector-spaces





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 6 at 21:53









Arnaud Mortier

19.2k22159




19.2k22159









asked Aug 6 at 21:03









Shayan Hemmati

263




263











  • I did. I used the exact same phrasing of the question from the online course I'm taking. I attached pictures to demonstrate my point.
    – Shayan Hemmati
    Aug 6 at 21:16










  • did the pictures provide any clarity or still a typo?
    – Shayan Hemmati
    Aug 6 at 21:19










  • Perfect! thank you for clarification
    – Shayan Hemmati
    Aug 6 at 21:23










  • Pjonin gave a good explanation, of course for the second one if we are dealing with square matrices of different size then "the set of all upper triangular square matrices is not a real vector space" is true of course!
    – gimusi
    Aug 6 at 21:28
















  • I did. I used the exact same phrasing of the question from the online course I'm taking. I attached pictures to demonstrate my point.
    – Shayan Hemmati
    Aug 6 at 21:16










  • did the pictures provide any clarity or still a typo?
    – Shayan Hemmati
    Aug 6 at 21:19










  • Perfect! thank you for clarification
    – Shayan Hemmati
    Aug 6 at 21:23










  • Pjonin gave a good explanation, of course for the second one if we are dealing with square matrices of different size then "the set of all upper triangular square matrices is not a real vector space" is true of course!
    – gimusi
    Aug 6 at 21:28















I did. I used the exact same phrasing of the question from the online course I'm taking. I attached pictures to demonstrate my point.
– Shayan Hemmati
Aug 6 at 21:16




I did. I used the exact same phrasing of the question from the online course I'm taking. I attached pictures to demonstrate my point.
– Shayan Hemmati
Aug 6 at 21:16












did the pictures provide any clarity or still a typo?
– Shayan Hemmati
Aug 6 at 21:19




did the pictures provide any clarity or still a typo?
– Shayan Hemmati
Aug 6 at 21:19












Perfect! thank you for clarification
– Shayan Hemmati
Aug 6 at 21:23




Perfect! thank you for clarification
– Shayan Hemmati
Aug 6 at 21:23












Pjonin gave a good explanation, of course for the second one if we are dealing with square matrices of different size then "the set of all upper triangular square matrices is not a real vector space" is true of course!
– gimusi
Aug 6 at 21:28




Pjonin gave a good explanation, of course for the second one if we are dealing with square matrices of different size then "the set of all upper triangular square matrices is not a real vector space" is true of course!
– gimusi
Aug 6 at 21:28










1 Answer
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I guess what they mean is that you need to precise that they must be the same size and that you can’t have a vectorial space of matrices of size p and of size n.



But that’s quite unclear.






share|cite|improve this answer





















  • Yes of course! I did'n consider that but that must be the reason.
    – gimusi
    Aug 6 at 21:26










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1 Answer
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
3
down vote













I guess what they mean is that you need to precise that they must be the same size and that you can’t have a vectorial space of matrices of size p and of size n.



But that’s quite unclear.






share|cite|improve this answer





















  • Yes of course! I did'n consider that but that must be the reason.
    – gimusi
    Aug 6 at 21:26














up vote
3
down vote













I guess what they mean is that you need to precise that they must be the same size and that you can’t have a vectorial space of matrices of size p and of size n.



But that’s quite unclear.






share|cite|improve this answer





















  • Yes of course! I did'n consider that but that must be the reason.
    – gimusi
    Aug 6 at 21:26












up vote
3
down vote










up vote
3
down vote









I guess what they mean is that you need to precise that they must be the same size and that you can’t have a vectorial space of matrices of size p and of size n.



But that’s quite unclear.






share|cite|improve this answer













I guess what they mean is that you need to precise that they must be the same size and that you can’t have a vectorial space of matrices of size p and of size n.



But that’s quite unclear.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Aug 6 at 21:25









Pjonin

3206




3206











  • Yes of course! I did'n consider that but that must be the reason.
    – gimusi
    Aug 6 at 21:26
















  • Yes of course! I did'n consider that but that must be the reason.
    – gimusi
    Aug 6 at 21:26















Yes of course! I did'n consider that but that must be the reason.
– gimusi
Aug 6 at 21:26




Yes of course! I did'n consider that but that must be the reason.
– gimusi
Aug 6 at 21:26












 

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