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Questions and Answers
What is a set of vectors said to be if the vector equation x1v1 + x2v2 + ... + xpvp = 0 has only trivial solutions?
What is a set of vectors said to be if the vector equation x1v1 + x2v2 + ... + xpvp = 0 has only trivial solutions?
Linearly independent
What does it mean if non-trivial solutions exist?
What does it mean if non-trivial solutions exist?
There are infinitely many solutions, indicating a dependent system.
What is a trivial solution?
What is a trivial solution?
The zero vector is a solution.
What is a pivot position in a matrix?
What is a pivot position in a matrix?
What does it mean for a matrix to be inconsistent?
What does it mean for a matrix to be inconsistent?
What is a consistent matrix?
What is a consistent matrix?
What indicates that a set of vectors is linearly dependent?
What indicates that a set of vectors is linearly dependent?
Are standard basis vectors linearly independent?
Are standard basis vectors linearly independent?
What does the Fundamental Theorem of Linear Algebra Part I state?
What does the Fundamental Theorem of Linear Algebra Part I state?
What is a basis for a vector space?
What is a basis for a vector space?
What is the span of vectors v1, v2,..., vn?
What is the span of vectors v1, v2,..., vn?
What makes a set of vectors linearly independent?
What makes a set of vectors linearly independent?
What is the dimension of the null space of a matrix A?
What is the dimension of the null space of a matrix A?
The dimensions of the column space of A are equal to the rank of A.
The dimensions of the column space of A are equal to the rank of A.
How do you find the basis set of a matrix A?
How do you find the basis set of a matrix A?
What defines an eigenvalue of a matrix?
What defines an eigenvalue of a matrix?
An n x n matrix A is invertible if and only if detA ≠0.
An n x n matrix A is invertible if and only if detA ≠0.
If a set of p vectors spans a p-dimensional subspace H of R^n, then these vectors form a basis for H.
If a set of p vectors spans a p-dimensional subspace H of R^n, then these vectors form a basis for H.
If H is a p-dimensional subspace of R^n, then a linearly independent set of p vectors in H is a basis for H.
If H is a p-dimensional subspace of R^n, then a linearly independent set of p vectors in H is a basis for H.
A matrix is invertible when the determinant is 0.
A matrix is invertible when the determinant is 0.
Three vectors, one of which is the zero vector, can form a basis for R^3.
Three vectors, one of which is the zero vector, can form a basis for R^3.
The only three-dimensional subspace of R^3 is R^3 itself.
The only three-dimensional subspace of R^3 is R^3 itself.
An n x n matrix A is diagonalizable if A has n distinct eigenvalues.
An n x n matrix A is diagonalizable if A has n distinct eigenvalues.
A set B={v1, v2,..., vn} of vectors is said to be an EIGENBASIS for R^n when there is an n x n matrix A such that B is a set of n linearly independent eigenvectors of A.
A set B={v1, v2,..., vn} of vectors is said to be an EIGENBASIS for R^n when there is an n x n matrix A such that B is a set of n linearly independent eigenvectors of A.
V1 and v2 are linearly independent eigenvectors of an n x n matrix A then they correspond to distinct eigenvalues of A.
V1 and v2 are linearly independent eigenvectors of an n x n matrix A then they correspond to distinct eigenvalues of A.
If eigenvectors of an n x n matrix A are a basis for R^n, then A is diagonalizable.
If eigenvectors of an n x n matrix A are a basis for R^n, then A is diagonalizable.
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Study Notes
Linear Independence and Solutions
- A set of vectors is linearly independent if the equation x1v1 + x2v2 + ... + xpvp = 0 has only the trivial solution (all constants are zero).
- Non-trivial solutions indicate the existence of infinitely many solutions and the presence of a free variable, suggesting the system is dependent.
- Trivial solutions refer to the zero vector being a solution in the vector equation.
Matrix Properties
- Pivot positions in a matrix occur where after row reduction, a leading one is found.
- An inconsistent matrix has no solutions, while a consistent matrix has at least one solution.
- A matrix is singular (not invertible) when its determinant equals zero.
Vector Spaces and Basis
- A basis for a vector space is a set of linearly independent vectors that spans the space.
- The span of vectors includes all possible linear combinations of those vectors.
- The standard basis vectors for R^n are linearly independent and correspond to the identity matrix.
Dimension and the Fundamental Theorem
- The dimensions of the column space (Col(A)) and null space (Nul(A)) of a matrix A satisfy the equation: dim(Col(A)) + dim(Nul(A)) = n.
- The number of pivot columns from the reduced row echelon form provides a basis for Col(A), while free variables reveal the basis for Nul(A).
Eigenvalues and Diagonalization
- An eigenvalue (λ) of a matrix A is a scalar such that there exists a non-trivial solution to Ax = λx; corresponding non-trivial solutions are the eigenvectors.
- A matrix is diagonalizable if it has n linearly independent eigenvectors or can be represented as A = PDP^(-1) with D being a diagonal matrix.
Theorems and Properties
- The dimensions of the column space equal the rank of A.
- If a set of p vectors spans a p-dimensional subspace, they form a basis for that subspace.
- A set is linearly dependent if it contains more vectors than dimensions (p > n) or if it contains the zero vector.
Miscellaneous
- Geometric multiplicity of an eigenvalue is the dimension of the null space of A - λI, while algebraic multiplicity is the number of times λ appears as a root of det(A - λI) = 0.
- A system of equations has unique solutions represented in terms of a basis when those solutions lie within a defined subspace.
Truth Statements for Review
- A matrix A is invertible if its determinant is non-zero.
- An eigenbasis consists of n linearly independent vectors corresponding to eigenvalues of a matrix.
- The projection of a vector y on a subspace W remains within W's orthogonal complement (W⊥).
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