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