APMA 3080 Final Exam Question and
answers already passed 2025/2026
How 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
answers already passed 2025/2026
How 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
