Elementary Linear Algebra Terms and
Theorems - David Lay Complete Latest
2025/2026 with Correct Answers and
Rationales GRADED A+
Let A be an m×n matrix. Then the following are equivalent:
(a) For each b in ℝᵐ, the equation Ax = b has a solution.
Theorem (1.4) (b) Each b in ℝᵐ is a linear combination of the columns of
A. (c) The columns of A span ℝᵐ. (d) A has a pivot
position in every row.
An indexed set S = {v₁, v₂, ..., vₚ} of two or more vectors is
Theorem (1.7) linearly dependent if and only if at least one vector is a
linear combination of the others.
If a set S = {v₁, v₂, ..., vₚ} in ℝⁿ contains the zero vector,
Theorem (1.9)
then it is linearly dependent.
Let T: ℝⁿ → ℝᵐ be a linear transformation. Then there
exists a unique matrix A such that T(x) = Ax for all x in ℝⁿ,
Theorem (1.10)
where A's jᵗʰ column is T(eⱼ) (with eⱼ as the jᵗʰ standard
basis vector).
Let T: ℝᵐ → ℝⁿ be a linear transformation. Then T is one-
Theorem (1.11) to-one if and only if the equation Ax = 0 has only the trivial
solution.
, For T: ℝᵐ → ℝⁿ with standard matrix A: (a) T maps ℝᵐ
onto ℝⁿ if and only if the columns of A span ℝᵐ. (b) T is
Theorem (1.12)
one-to-one if and only if the columns of A are linearly
independent.
For a square n×n matrix A, the following are equivalent:
(a) A is invertible. (b) A is row equivalent to Iₙ. (c) A has n
pivot positions. (d) Ax = 0 has only the trivial solution. (e)
A's columns are linearly independent. (f) The linear
Theorem (2.8) transformation x → Ax is one-to-one. (g) Ax = b has a
solution for every b in ℝⁿ. (h) A's columns span ℝⁿ. (i) x →
Ax maps ℝⁿ onto ℝⁿ. (j) There exists an n×n matrix C such
that CA = I. (k) There exists an n×n matrix D such that AD
= I. (l) Aᵀ is invertible.
A matrix A is invertible if and only if its columns form a
Invertible Matrix Definition
basis for ℝⁿ.
A matrix A is row equivalent to Iₙ if it can be reduced to Iₙ
Row Equivalent Definition
using elementary row operations.
If A is invertible, there exist matrices C and D such that CA
Invertible Matrix = I and AD = I. These matrices are both equal to A⁻¹, the
unique inverse of A.
Transpose of Invertible If A is invertible, then Aᵀ is also invertible. The inverse of
Matrix Aᵀ is (A⁻¹)ᵀ, ensuring the invertibility of Aᵀ.
The equation Ax = b has a solution for every b in ℝⁿ if and
only if the columns of A span ℝⁿ. A's columns spanning ℝⁿ
Solution of Ax = b
guarantees that every vector b can be expressed as a
combination of them.
Theorems - David Lay Complete Latest
2025/2026 with Correct Answers and
Rationales GRADED A+
Let A be an m×n matrix. Then the following are equivalent:
(a) For each b in ℝᵐ, the equation Ax = b has a solution.
Theorem (1.4) (b) Each b in ℝᵐ is a linear combination of the columns of
A. (c) The columns of A span ℝᵐ. (d) A has a pivot
position in every row.
An indexed set S = {v₁, v₂, ..., vₚ} of two or more vectors is
Theorem (1.7) linearly dependent if and only if at least one vector is a
linear combination of the others.
If a set S = {v₁, v₂, ..., vₚ} in ℝⁿ contains the zero vector,
Theorem (1.9)
then it is linearly dependent.
Let T: ℝⁿ → ℝᵐ be a linear transformation. Then there
exists a unique matrix A such that T(x) = Ax for all x in ℝⁿ,
Theorem (1.10)
where A's jᵗʰ column is T(eⱼ) (with eⱼ as the jᵗʰ standard
basis vector).
Let T: ℝᵐ → ℝⁿ be a linear transformation. Then T is one-
Theorem (1.11) to-one if and only if the equation Ax = 0 has only the trivial
solution.
, For T: ℝᵐ → ℝⁿ with standard matrix A: (a) T maps ℝᵐ
onto ℝⁿ if and only if the columns of A span ℝᵐ. (b) T is
Theorem (1.12)
one-to-one if and only if the columns of A are linearly
independent.
For a square n×n matrix A, the following are equivalent:
(a) A is invertible. (b) A is row equivalent to Iₙ. (c) A has n
pivot positions. (d) Ax = 0 has only the trivial solution. (e)
A's columns are linearly independent. (f) The linear
Theorem (2.8) transformation x → Ax is one-to-one. (g) Ax = b has a
solution for every b in ℝⁿ. (h) A's columns span ℝⁿ. (i) x →
Ax maps ℝⁿ onto ℝⁿ. (j) There exists an n×n matrix C such
that CA = I. (k) There exists an n×n matrix D such that AD
= I. (l) Aᵀ is invertible.
A matrix A is invertible if and only if its columns form a
Invertible Matrix Definition
basis for ℝⁿ.
A matrix A is row equivalent to Iₙ if it can be reduced to Iₙ
Row Equivalent Definition
using elementary row operations.
If A is invertible, there exist matrices C and D such that CA
Invertible Matrix = I and AD = I. These matrices are both equal to A⁻¹, the
unique inverse of A.
Transpose of Invertible If A is invertible, then Aᵀ is also invertible. The inverse of
Matrix Aᵀ is (A⁻¹)ᵀ, ensuring the invertibility of Aᵀ.
The equation Ax = b has a solution for every b in ℝⁿ if and
only if the columns of A span ℝⁿ. A's columns spanning ℝⁿ
Solution of Ax = b
guarantees that every vector b can be expressed as a
combination of them.
