APMA 3080 - Final Exam Review Questions and Answers Latest Update 2025 Rated 100%
Span - Answers The set of all linear combinations x₁u₁ + ... + x ₙu ₙ, where x₁, ..., x ₙ can be any real
numbers.
Linear Independence - Answers The only solution to the vector equation x₁u₁ + ... + x ₙu ₙ = 0 is the trivial
solution.
Linearly Dependent - Answers If a set of vectors contains the zero vector, is the set linearly dependent or
independent?
Linearly Dependent - Answers If an nxm set of vectors in Rⁿ exists such n < m, is the set linearly
dependent or independent?
Linearly Dependent - Answers If one of the vectors in a set of vectors is a linear combination of one of
the other vectors, is the set linearly dependent or independent?
Ax = {0} - Answers General Form of a Homogeneous Linear System
1. Closed under addition, 2. Closed under scalar multiplication - Answers Conditions Required to Form a
Transformation
One-to-One - Answers Let T be a linear transformation defined by T(x) = Ax. The columns of A are
linearly independent.
Onto - Answers Let T be a linear transformation defined by T(x) = Ax. The columns of A span Rⁿ.
One-to-One - Answers Let T be a linear transformation. T(x) ={0} has only the trivial solution x = {0}.
1. Contains the zero vector, 2. Closed under addition, 3. Closed under scalar multiplication - Answers
Conditions Required to Form a Subspace
Yes - Answers Is a span a subspace?
Null Space - Answers The set of solutions to the homogeneous linear system Ax = {0}, where A is an nxm
matrix.
Kernel - Answers A subspace of the domain of a linear transformation T.
Range - Answers A subspace of the codomain of a linear transformation T.
Kernel - Answers The set of all vectors x such that T(x) = {0}, where T is a linear transformation.
1. Spans the subspace, 2. Linearly independent - Answers Conditions Required to Form a Basis
No - Answers If the number of vectors in a set in a subspace is less than the dimension of the subspace,
does the set span the subspace?
Span - Answers The set of all linear combinations x₁u₁ + ... + x ₙu ₙ, where x₁, ..., x ₙ can be any real
numbers.
Linear Independence - Answers The only solution to the vector equation x₁u₁ + ... + x ₙu ₙ = 0 is the trivial
solution.
Linearly Dependent - Answers If a set of vectors contains the zero vector, is the set linearly dependent or
independent?
Linearly Dependent - Answers If an nxm set of vectors in Rⁿ exists such n < m, is the set linearly
dependent or independent?
Linearly Dependent - Answers If one of the vectors in a set of vectors is a linear combination of one of
the other vectors, is the set linearly dependent or independent?
Ax = {0} - Answers General Form of a Homogeneous Linear System
1. Closed under addition, 2. Closed under scalar multiplication - Answers Conditions Required to Form a
Transformation
One-to-One - Answers Let T be a linear transformation defined by T(x) = Ax. The columns of A are
linearly independent.
Onto - Answers Let T be a linear transformation defined by T(x) = Ax. The columns of A span Rⁿ.
One-to-One - Answers Let T be a linear transformation. T(x) ={0} has only the trivial solution x = {0}.
1. Contains the zero vector, 2. Closed under addition, 3. Closed under scalar multiplication - Answers
Conditions Required to Form a Subspace
Yes - Answers Is a span a subspace?
Null Space - Answers The set of solutions to the homogeneous linear system Ax = {0}, where A is an nxm
matrix.
Kernel - Answers A subspace of the domain of a linear transformation T.
Range - Answers A subspace of the codomain of a linear transformation T.
Kernel - Answers The set of all vectors x such that T(x) = {0}, where T is a linear transformation.
1. Spans the subspace, 2. Linearly independent - Answers Conditions Required to Form a Basis
No - Answers If the number of vectors in a set in a subspace is less than the dimension of the subspace,
does the set span the subspace?
