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What is the dimension of a square matrix A?
What is the dimension of a square matrix A?
Dim(A) = n × n
If T_A: Rⁿ → Rⁿ is multiplication by A, then this statement is considered a definition or proof.
If T_A: Rⁿ → Rⁿ is multiplication by A, then this statement is considered a definition or proof.
True (A)
If A is invertible, then A⁻¹A equals the identity matrix.
If A is invertible, then A⁻¹A equals the identity matrix.
True (A)
The equation Ax = 0 has only the trivial solution if A is not invertible.
The equation Ax = 0 has only the trivial solution if A is not invertible.
The reduced row-echelon form of A is I(n) if A is invertible.
The reduced row-echelon form of A is I(n) if A is invertible.
A is expressible as a product of elementary matrices if A is invertible.
A is expressible as a product of elementary matrices if A is invertible.
The equation Ax = b is consistent for every n×1 matrix b if A is invertible.
The equation Ax = b is consistent for every n×1 matrix b if A is invertible.
The equation ax = b has exactly one solution for every n×1 matrix b if A is invertible.
The equation ax = b has exactly one solution for every n×1 matrix b if A is invertible.
Det(A) ≠ 0 if A is invertible.
Det(A) ≠ 0 if A is invertible.
The range of T_A is Rⁿ if A is invertible.
The range of T_A is Rⁿ if A is invertible.
T_A is one-to-one if A is invertible.
T_A is one-to-one if A is invertible.
The column vectors of A are linearly independent if the rank of A equals n.
The column vectors of A are linearly independent if the rank of A equals n.
The row vectors of A are linearly independent if the rank of A equals n.
The row vectors of A are linearly independent if the rank of A equals n.
The column vectors of A span Rⁿ if A is invertible.
The column vectors of A span Rⁿ if A is invertible.
The row vectors of A span Rⁿ if A is invertible.
The row vectors of A span Rⁿ if A is invertible.
The column vectors of A form a basis for Rⁿ if A is invertible.
The column vectors of A form a basis for Rⁿ if A is invertible.
The row vectors of A form a basis for Rⁿ if A is invertible.
The row vectors of A form a basis for Rⁿ if A is invertible.
A has rank n if it is an n × n matrix.
A has rank n if it is an n × n matrix.
A has nullity 0 if it is invertible.
A has nullity 0 if it is invertible.
The orthogonal complement of the nullspace of A is equal to Rⁿ if A is invertible.
The orthogonal complement of the nullspace of A is equal to Rⁿ if A is invertible.
The orthogonal complement of the row space of A is {0} if A is invertible.
The orthogonal complement of the row space of A is {0} if A is invertible.
(A^T)A is invertible if A is invertible.
(A^T)A is invertible if A is invertible.
λ = 0 is not an eigenvalue of A if A is invertible.
λ = 0 is not an eigenvalue of A if A is invertible.
A is row equivalent to the n × n identity matrix if it has n pivot positions.
A is row equivalent to the n × n identity matrix if it has n pivot positions.
A has n pivot positions if the reduced row-echelon form of A contains leading 1's.
A has n pivot positions if the reduced row-echelon form of A contains leading 1's.
The equation Ax = 0 has only the trivial solution if A has full rank.
The equation Ax = 0 has only the trivial solution if A has full rank.
The equation Ax = b has at least one solution for each b in Rn if A is invertible.
The equation Ax = b has at least one solution for each b in Rn if A is invertible.
The columns of A span Rn if A is not invertible.
The columns of A span Rn if A is not invertible.
The linear transformation x → Ax maps Rn onto Rn if A is invertible.
The linear transformation x → Ax maps Rn onto Rn if A is invertible.
There is an n × n matrix C such that CA = I if A is invertible.
There is an n × n matrix C such that CA = I if A is invertible.
There is an n × n matrix D such that AD = I if A is invertible.
There is an n × n matrix D such that AD = I if A is invertible.
The columns of A form a basis of Rn if A is invertible.
The columns of A form a basis of Rn if A is invertible.
What does 'redech' refer to?
What does 'redech' refer to?
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Study Notes
Matrix Definitions and Properties
- A square matrix A is defined by dimensions: Dim(A) = n×n, indicating it has n rows and n columns.
- Multiplication by matrix A is represented as T_A: Rⁿ → Rⁿ.
Invertibility of Matrix A
- A is invertible if A⁻¹A equals the identity matrix, confirming the existence of an inverse.
- If the equation Ax = 0 has only the trivial solution (x = 0), A is guaranteed to be invertible.
Row and Column Echelon Forms
- The reduced row-echelon form (RREF) of A is equal to I(n), the identity matrix, confirming A's invertibility.
- Expressibility of A as a product of elementary matrices implies that it can be transformed into the identity matrix through row operations.
Consistency and Solutions of Linear Equations
- The equation Ax = b is consistent for every n×1 matrix b, ensuring that solutions exist for all vectors in Rⁿ.
- The equation ax = b has exactly one solution for every n×1 matrix b, further confirming the behavior of linear transformations.
Determinants and Ranges
- A non-zero determinant (det(A) ≠ 0) confirms that matrix A is invertible.
- The range of T_A being equal to Rⁿ signifies that all possible output vectors can be achieved through multiplication by A.
One-to-One Transformations
- T_A is one-to-one if distinct input vectors lead to distinct output vectors, ensuring uniqueness in solutions.
- The column vectors of A are linearly independent if Rrow(A) contains all non-zero columns, highlighting a lack of redundancy among columns.
Spanning and Bases
- The row vectors of A are linearly independent if Rrow(A) holds all non-zero rows, indicating no linear combinations can lead to a nontrivial zero vector.
- The column vectors span Rⁿ if they can generate every vector in Rⁿ through linear combinations.
- Row vectors of A also span Rⁿ if they can similarly cover the space through combinations.
Basis and Rank
- The column vectors of A form a basis for Rⁿ when they are linearly independent and span the entire space.
- The row vectors form a basis for Rⁿ under the same linear independence and spanning criteria.
- Matrix A has rank n if it contains n pivot positions, meaning all rows (or columns) are linearly independent.
- A has nullity 0 if the null space contains only the zero vector, confirming that Ax = 0 only has the trivial solution.
Orthogonal Complements
- The orthogonal complement of the nullspace of A equates to Rⁿ, indicating complete coverage of the vector space.
- The orthogonal complement of the row space of A is {0}, reinforcing the notion of linear independence.
Invertibility of Product Matrices
- (A^T)A is invertible if A is invertible, providing a relationship between a matrix and its transpose.
Eigenvalues
- If λ = 0 is not an eigenvalue of A, then A is guaranteed to be invertible, impacting solutions and behaviors in linear transformations.
Equivalence and Relationships
- A is row equivalent to the n×n identity matrix if it can be transformed into it through a series of elementary row operations.
- Matrix A has n pivot positions, ensuring that each column and row in the RREF of A is non-zero.
- Collected conditions guarantee that the columns of A form a basis of Rⁿ, providing full representation of the vector space.
Additional Concepts
- "Redech" refers to the reduced row-echelon form, which plays a vital role in analyzing matrix properties and their implications in linear algebra.
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