Questions and Answers
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?
There are infinitely many solutions, indicating a dependent system.
What is a trivial solution?
The zero vector is a solution.
What is a pivot position in a matrix?
Signup and view all the answers
What does it mean for a matrix to be inconsistent?
Signup and view all the answers
What is a consistent matrix?
Signup and view all the answers
What indicates that a set of vectors is linearly dependent?
Signup and view all the answers
Are standard basis vectors linearly independent?
Signup and view all the answers
What does the Fundamental Theorem of Linear Algebra Part I state?
Signup and view all the answers
What is a basis for a vector space?
Signup and view all the answers
What is the span of vectors v1, v2,..., vn?
Signup and view all the answers
What makes a set of vectors linearly independent?
Signup and view all the answers
What is the dimension of the null space of a matrix A?
Signup and view all the answers
The dimensions of the column space of A are equal to the rank of A.
Signup and view all the answers
How do you find the basis set of a matrix A?
Signup and view all the answers
What defines an eigenvalue of a matrix?
Signup and view all the answers
An n x n matrix A is invertible if and only if detA ≠ 0.
Signup and view all the answers
If a set of p vectors spans a p-dimensional subspace H of R^n, then these vectors form a basis for H.
Signup and view all the answers
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.
Signup and view all the answers
A matrix is invertible when the determinant is 0.
Signup and view all the answers
Three vectors, one of which is the zero vector, can form a basis for R^3.
Signup and view all the answers
The only three-dimensional subspace of R^3 is R^3 itself.
Signup and view all the answers
An n x n matrix A is diagonalizable if A has n distinct eigenvalues.
Signup and view all the answers
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.
Signup and view all the answers
V1 and v2 are linearly independent eigenvectors of an n x n matrix A then they correspond to distinct eigenvalues of A.
Signup and view all the answers
If eigenvectors of an n x n matrix A are a basis for R^n, then A is diagonalizable.
Signup and view all the answers
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⊥).
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge with these flashcards covering key concepts from Linear Algebra Exam 3. Learn about linear independence and the implications of trivial versus non-trivial solutions in vector equations. Perfect for reviewing before the exam!
