MATH 208 CERTIFICATION SCRIPT 2026
QUESTIONS WITH SOLUTIONS GRADED A+
◍ What is the Row Space?.
Answer: Row(A) = span of rows; Basis = nonzero rows of RREF.
◍ What is a Subspace?.
Answer: 1) Contains 0; 2) Closed under addition; 3) Closed under scalar
multiplication.
◍ How do you find the Null Space (Step-by-Step)?.
Answer: 1) Solve Ax=0; 2) Row reduce A; 3) Identify free variables; 4)
Write parametric solution; 5) Extract basis vectors; Free variables = Nullity.
◍ (sets).
Answer: empty set, also written using the roster style as { }
◍ DeMorgan's Law ¬(p q).
Answer: (¬p ¬q)
◍ How do you find a Basis using the Column Method?.
Answer: Put vectors as columns; Row reduce; Pivot columns → original
columns form basis.
◍ A ∩ B (sets).
Answer: Where A shares with B
◍ What is Dimension?.
Answer: Number of vectors in a basis; R^n has dimension n; {0} has
dimension 0.
◍ What is the rule for Matrix Multiplication?.
Answer: A(n×m) * B(m×p) exists; Inner numbers must match; Not
commutative.
, ◍ equivalence class.
Answer: Suppose E is an equivalence relation on a set A and that x is one
particular element of A. The equivalence class of x is the set of all the things
in A that are equivalent to x.For instance, think once more about the deck of
cards with the equivalence relation having the same rank. The equivalence
class of the two of spades would be the set [2♠] = {2♣, 2 , 2 , 2♠}. That
would also be the equivalence class of the two of diamonds. On the other
hand, if the equivalence relation we are using for the deck is having the
same suit, then the equivalence class of the two of spades would be [2♠] =
{A♠, 2♠, 3♠, 4♠, 5♠, 6♠, 7♠, 8♠, 9♠, 10♠, J♠, Q♠, K♠}.
◍ Disjunctive Syllogism p q and ¬p.
Answer: q
◍ What is a Linear Transformation?.
Answer: T: R^m -> R^n satisfying: 1) T(u+v) = T(u) + T(v) 2) T(ru) = rT(u)
◍ What is the Kernel of T?.
Answer: ker(T) = {x : T(x)=0}; Subspace of domain; T one-to-one iff
ker(T)={0}.
◍ a^(b*c) =.
Answer: (a^b)^c
◍ codomain.
Answer: G(x, y) the set B, the domain of discourse for Y
◍ What is a Basis?.
Answer: Spans subspace AND is linearly independent; Smallest spanning
set.
◍ list of logical connectives (not, etc).
Answer: OR ( )AND ( )Negation/ NOT (¬)Implication / if-then (→)If and
only if ( ).exclusive or (xor, )
◍ 0-1 matrix.
Answer: look in notes
QUESTIONS WITH SOLUTIONS GRADED A+
◍ What is the Row Space?.
Answer: Row(A) = span of rows; Basis = nonzero rows of RREF.
◍ What is a Subspace?.
Answer: 1) Contains 0; 2) Closed under addition; 3) Closed under scalar
multiplication.
◍ How do you find the Null Space (Step-by-Step)?.
Answer: 1) Solve Ax=0; 2) Row reduce A; 3) Identify free variables; 4)
Write parametric solution; 5) Extract basis vectors; Free variables = Nullity.
◍ (sets).
Answer: empty set, also written using the roster style as { }
◍ DeMorgan's Law ¬(p q).
Answer: (¬p ¬q)
◍ How do you find a Basis using the Column Method?.
Answer: Put vectors as columns; Row reduce; Pivot columns → original
columns form basis.
◍ A ∩ B (sets).
Answer: Where A shares with B
◍ What is Dimension?.
Answer: Number of vectors in a basis; R^n has dimension n; {0} has
dimension 0.
◍ What is the rule for Matrix Multiplication?.
Answer: A(n×m) * B(m×p) exists; Inner numbers must match; Not
commutative.
, ◍ equivalence class.
Answer: Suppose E is an equivalence relation on a set A and that x is one
particular element of A. The equivalence class of x is the set of all the things
in A that are equivalent to x.For instance, think once more about the deck of
cards with the equivalence relation having the same rank. The equivalence
class of the two of spades would be the set [2♠] = {2♣, 2 , 2 , 2♠}. That
would also be the equivalence class of the two of diamonds. On the other
hand, if the equivalence relation we are using for the deck is having the
same suit, then the equivalence class of the two of spades would be [2♠] =
{A♠, 2♠, 3♠, 4♠, 5♠, 6♠, 7♠, 8♠, 9♠, 10♠, J♠, Q♠, K♠}.
◍ Disjunctive Syllogism p q and ¬p.
Answer: q
◍ What is a Linear Transformation?.
Answer: T: R^m -> R^n satisfying: 1) T(u+v) = T(u) + T(v) 2) T(ru) = rT(u)
◍ What is the Kernel of T?.
Answer: ker(T) = {x : T(x)=0}; Subspace of domain; T one-to-one iff
ker(T)={0}.
◍ a^(b*c) =.
Answer: (a^b)^c
◍ codomain.
Answer: G(x, y) the set B, the domain of discourse for Y
◍ What is a Basis?.
Answer: Spans subspace AND is linearly independent; Smallest spanning
set.
◍ list of logical connectives (not, etc).
Answer: OR ( )AND ( )Negation/ NOT (¬)Implication / if-then (→)If and
only if ( ).exclusive or (xor, )
◍ 0-1 matrix.
Answer: look in notes
