APMA 3080 - Final Exam Review
2025/2026 Exam Questions and Answers
| 100% Solved
Span - 🧠 ANSWER ✔✔The set of all linear combinations x₁u₁ + ... + xₙuₙ,
where x₁, ..., xₙ can be any real numbers.
Linear Independence - 🧠 ANSWER ✔✔The only solution to the vector
equation x₁u₁ + ... + xₙuₙ = 0 is the trivial solution.
Linearly Dependent - 🧠 ANSWER ✔✔If a set of vectors contains the zero
vector, is the set linearly dependent or independent?
Linearly Dependent - 🧠 ANSWER ✔✔If an nxm set of vectors in Rⁿ exists
such n < m, is the set linearly dependent or independent?
Linearly Dependent - 🧠 ANSWER ✔✔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} - 🧠 ANSWER ✔✔General Form of a Homogeneous Linear System
1
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, 1. Closed under addition, 2. Closed under scalar multiplication - 🧠
ANSWER ✔✔Conditions Required to Form a Transformation
One-to-One - 🧠 ANSWER ✔✔Let T be a linear transformation defined by
T(x) = Ax. The columns of A are linearly independent.
Onto - 🧠 ANSWER ✔✔Let T be a linear transformation defined by T(x) =
Ax. The columns of A span Rⁿ.
One-to-One - 🧠 ANSWER ✔✔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 - 🧠 ANSWER ✔✔Conditions Required to Form a
Subspace
Yes - 🧠 ANSWER ✔✔Is a span a subspace?
Null Space - 🧠 ANSWER ✔✔The set of solutions to the homogeneous
linear system Ax = {0}, where A is an nxm matrix.
Kernel - 🧠 ANSWER ✔✔A subspace of the domain of a linear
transformation T.
2
COPYRIGHT@SARAHROSAPERAL 2025/2026. YEAR PUBLISHED 2025. COMPANY REGISTRATION NUMBER: 619652435. TERMS OF USE.
PRIVACY STATEMENT. ALL RIGHTS RESERVED
2025/2026 Exam Questions and Answers
| 100% Solved
Span - 🧠 ANSWER ✔✔The set of all linear combinations x₁u₁ + ... + xₙuₙ,
where x₁, ..., xₙ can be any real numbers.
Linear Independence - 🧠 ANSWER ✔✔The only solution to the vector
equation x₁u₁ + ... + xₙuₙ = 0 is the trivial solution.
Linearly Dependent - 🧠 ANSWER ✔✔If a set of vectors contains the zero
vector, is the set linearly dependent or independent?
Linearly Dependent - 🧠 ANSWER ✔✔If an nxm set of vectors in Rⁿ exists
such n < m, is the set linearly dependent or independent?
Linearly Dependent - 🧠 ANSWER ✔✔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} - 🧠 ANSWER ✔✔General Form of a Homogeneous Linear System
1
COPYRIGHT@SARAHROSAPERAL 2025/2026. YEAR PUBLISHED 2025. COMPANY REGISTRATION NUMBER: 619652435. TERMS OF USE.
PRIVACY STATEMENT. ALL RIGHTS RESERVED
, 1. Closed under addition, 2. Closed under scalar multiplication - 🧠
ANSWER ✔✔Conditions Required to Form a Transformation
One-to-One - 🧠 ANSWER ✔✔Let T be a linear transformation defined by
T(x) = Ax. The columns of A are linearly independent.
Onto - 🧠 ANSWER ✔✔Let T be a linear transformation defined by T(x) =
Ax. The columns of A span Rⁿ.
One-to-One - 🧠 ANSWER ✔✔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 - 🧠 ANSWER ✔✔Conditions Required to Form a
Subspace
Yes - 🧠 ANSWER ✔✔Is a span a subspace?
Null Space - 🧠 ANSWER ✔✔The set of solutions to the homogeneous
linear system Ax = {0}, where A is an nxm matrix.
Kernel - 🧠 ANSWER ✔✔A subspace of the domain of a linear
transformation T.
2
COPYRIGHT@SARAHROSAPERAL 2025/2026. YEAR PUBLISHED 2025. COMPANY REGISTRATION NUMBER: 619652435. TERMS OF USE.
PRIVACY STATEMENT. ALL RIGHTS RESERVED
