Questions and Answers
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.
True
If A is invertible, then A⁻¹A equals the identity matrix.
True
The equation Ax = 0 has only the trivial solution if A is not invertible.
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The reduced row-echelon form of A is I(n) if A is invertible.
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A is expressible as a product of elementary matrices if A is invertible.
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The equation Ax = b is consistent for every n×1 matrix b if A is invertible.
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The equation ax = b has exactly one solution for every n×1 matrix b if A is invertible.
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Det(A) ≠ 0 if A is invertible.
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The range of T_A is Rⁿ if A is invertible.
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T_A is one-to-one if A is invertible.
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The column vectors of A are linearly independent if the rank of A equals n.
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The row vectors of A are linearly independent if the rank of A equals n.
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The column vectors of A span Rⁿ if A is invertible.
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The row vectors of A span Rⁿ if A is invertible.
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The column vectors of A form a basis for Rⁿ if A is invertible.
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The row vectors of A form a basis for Rⁿ if A is invertible.
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A has rank n if it is an n × n matrix.
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A has nullity 0 if it is invertible.
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The orthogonal complement of the nullspace of A is equal to Rⁿ if A is invertible.
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The orthogonal complement of the row space of A is {0} if A is invertible.
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(A^T)A is invertible if A is invertible.
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λ = 0 is not an eigenvalue of A if A is invertible.
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A is row equivalent to the n × n identity matrix if it has n pivot positions.
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A has n pivot positions if the reduced row-echelon form of A contains leading 1's.
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The equation Ax = 0 has only the trivial solution if A has full rank.
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The equation Ax = b has at least one solution for each b in Rn if A is invertible.
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The columns of A span Rn if A is not invertible.
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The linear transformation x → Ax maps Rn onto Rn if A is invertible.
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There is an n × n matrix C such that CA = I if A is invertible.
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There is an n × n matrix D such that AD = I if A is invertible.
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The columns of A form a basis of Rn if A is invertible.
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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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Description
Test your understanding of key concepts in Linear Algebra with these flashcards focusing on equivalent statements for square matrices and their properties. This quiz covers definitions and proofs of invertibility, solutions of equations, and reduced row-echelon forms.
