Exam (elaborations)
APMA 3080 Final Exam questions well answered
- Course
- Institution
APMA 3080 Final Exam questions well answered
[Show more]
Content preview
APMA 3080 Final ExamHow to show a set of vectors are linearly independent? - correct answer ✔✔(1) B = RREF([A | 0])
(2) Check if the only answer for B is x1, x2, x3... = 0
(3) If is, it is linearly independent. Otherwise, no.
difference between span, subspace and basis - correct answer ✔✔Subspace and span are linear
combinations of vectors. They contain infinitely many vectors. THEY ARE THE SAME!
A basis is a only set of vectors.
How to show subset S is a subspace? - correct answer ✔✔Check 3 conditions:
(1) if 0 is a part of S.
(2) if closed under addition: if (u in S) and (v in S), then (u + v in S).
(3) if closed under multiplication: if (u in S), then (cu in S).
Shortcuts:
- If S is a null space OR S is a span, it is automatically a subspace.
How to check if a set of vectors A form a basis of a subspace S? - correct answer ✔✔Check 2 conditions:
(1) A spans S. (That is, A can generate every vector in S.)
(2) A is linearly independent
How to find basis for a space? - correct answer ✔✔Method 1: Find basis for S = span(u1, u2, u3)
(1) Use the vectors {u1, u2, u3} to form rows of a matrix A
(2) B = REF(A)
(3) The non-zero rows of B are the basis
Method 1: Find basis for S = span(u1, u2, u3)
, (1) Use the vectors {u1, u2, u3} to form columns of a matrix A
(2) B = REF(A)
(3) pB = pivot columns of B
(4) Columns of A corresponding to pB form a basis for S.
How to find dimension of subspace - correct answer ✔✔(1) Find the number of free variables. Example:
dim([a b 0]) == 2
(2) OR, Count the number of vectors in the subspace's basis
Can a A subspace can have multiple bases? - correct answer ✔✔Yes
row space - correct answer ✔✔the set of all linear combinations of the row vectors
Column Space - correct answer ✔✔the set of all linear combinations of the columns of A
How to find basis for a span of vectors - correct answer ✔✔TODO
rank(A) - correct answer ✔✔length(pivot columns)
Rank-Nullity Theorem - correct answer ✔✔rank(A) + nullity(A) = number of columns
How to find Eigen-values? - correct answer ✔✔det(A - lambda*I) = 0
How to find Eigen-space? - correct answer ✔✔Null(A - lambda*I)
How to find char ploy? - correct answer ✔✔P(lambda) = det(A - lambda*I)
How to find the maximum set of Linear independent eigen vectors? - correct answer ✔✔(1) Find eigen
values
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller Sakayobako30. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $11.49. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
75632 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now