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1) Let's start with 2x2 matrices. We have addition and scalar multiplication defined for these objects. Explain how to do both for these objects to someone who is new to matrices. 2) Do we have closure under these operations? Explain what closure under the operations means and then explain why/why not each operation is/is not closed.

1) Let's start with 2x2 matrices. We have addition and scalar multiplication defined for these objects. Explain how to do both for these objects to someone who is new to matrices. 2) Do we have closure under these operations? Explain what closure under the operations means and then explain why/why not each operation is/is not closed.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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1) Let's start with 2x2 matrices. We have addition and scalar multiplication defined for these objects. Explain how to do both for these objects to someone who is new to matrices. 2) Do we have closure under these operations? Explain what closure under the operations means and then explain why/why not each operation is/is not closed. 3) Does it seem reasonable that the 8 axioms will hold for these objects? If not find one that you think will not work and show it does not work (you an use an actual example with numbers). If so pick 2 of the axioms and explain how they hold for the objects (do not use numbers in your matrices- these should hold for ALL 2x2 matrices remember). 4) Given your work in 1-3 would you say the set of all 2x2 matrices forms a vector space? Explain. 5) Given your work above do you think the set of all mxn matrices will form a vector space? Explain. 6) We defined span and looked at the span of a set of vectors geometrically and numerically working with vectors from R". When a set of vectors spans a space the linear combination of them generates every object in the space. Can you think of a set of 2x2 matrices such that when you take the span of them you can generate any 2x2 matrix? (hint: think of our standard basis for R², R³, .. . etc.) 7) If you came up with a set of objects in #6 does the set form a basis for the set of all 2x2 matrices? Explain why you think both requirements for a basis are/are not met. 8) What other "objects" have you worked with in mathematics that could potentially form a vector space?
Transcribed Image Text:1) Let's start with 2x2 matrices. We have addition and scalar multiplication defined for these objects. Explain how to do both for these objects to someone who is new to matrices. 2) Do we have closure under these operations? Explain what closure under the operations means and then explain why/why not each operation is/is not closed. 3) Does it seem reasonable that the 8 axioms will hold for these objects? If not find one that you think will not work and show it does not work (you an use an actual example with numbers). If so pick 2 of the axioms and explain how they hold for the objects (do not use numbers in your matrices- these should hold for ALL 2x2 matrices remember). 4) Given your work in 1-3 would you say the set of all 2x2 matrices forms a vector space? Explain. 5) Given your work above do you think the set of all mxn matrices will form a vector space? Explain. 6) We defined span and looked at the span of a set of vectors geometrically and numerically working with vectors from R". When a set of vectors spans a space the linear combination of them generates every object in the space. Can you think of a set of 2x2 matrices such that when you take the span of them you can generate any 2x2 matrix? (hint: think of our standard basis for R², R³, .. . etc.) 7) If you came up with a set of objects in #6 does the set form a basis for the set of all 2x2 matrices? Explain why you think both requirements for a basis are/are not met. 8) What other "objects" have you worked with in mathematics that could potentially form a vector space?
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