Questions and Answers
Define the kernel of a transformation T that maps from the vector space U to the vector space U.
ker(T) is the set of vectors in U that when transformed equal the zero vector in V.
Define the image of a transformation T that maps from the vector space U to the vector space U.
im(T) is the set of all transformed vectors from U that are in V.
What can we say about the kernel and the image of T relative to the domain and codomain?
ker(T) is a subspace of U (domain), im(T) is a subspace of V (codomain).
How are the kernel and image of a linear transformation related to the range (column space) and null space of a matrix?
Signup and view all the answers
What is the null space of a matrix?
Signup and view all the answers
If T is a linear transformation, what is its nullity?
Signup and view all the answers
If T is a linear transformation, what is its rank?
Signup and view all the answers
State the rank-nullity theorem.
Signup and view all the answers
What does it mean for two nxn matrices, A and B, to be similar?
Signup and view all the answers
State the three parts of the equivalence relation of similarity.
Signup and view all the answers
What is the 'equivalence' that similar matrices share?
Signup and view all the answers
Define what it means for an nxn matrix A to be diagonalisable.
Signup and view all the answers
Define eigenvalues and eigenvectors of an nxn matrix A.
Signup and view all the answers
What is the one vector eigenvectors can never be?
Signup and view all the answers
How do we find eigenvalues?
Signup and view all the answers
What is the characteristic polynomial of an nxn matrix A?
Signup and view all the answers
How do we use the characteristic polynomial of an nxn matrix A to find its eigenvalues?
Signup and view all the answers
What is the trace of a square matrix A?
Signup and view all the answers
How do we work out the trace and determinant of a square matrix A given its characteristic polynomial?
Signup and view all the answers
When do two matrices have the same characteristic polynomial?
Signup and view all the answers
Do two matrices with the same characteristic polynomial imply that they are similar?
Signup and view all the answers
If A and B are similar matrices, what equivalent properties do they share?
Signup and view all the answers
What is the link between eigenvalues and the trace and determinant of a square matrix A?
Signup and view all the answers
Study Notes
Kernel and Image of a Transformation
- The kernel, ker(T), of a transformation T consists of vectors in the domain U that map to the zero vector in the codomain V.
- The image, im(T), of a transformation T is the set of all vectors in V that are outputs of vectors from U.
Subspaces of Kernel and Image
- The kernel is a subspace of the domain U.
- The image of the transformation is a subspace of the codomain V.
Null Space of a Matrix
- The null space is defined as the set of solutions to the equation Ax = 0, providing insight into the kernel of the associated linear transformation.
Nullity and Rank
- Nullity of a linear transformation T is the dimension of its kernel, indicating how many vectors map to zero.
- Rank of T represents the dimension of its image, reflecting the number of linearly independent outputs.
Rank-Nullity Theorem
- The rank-nullity theorem establishes that the sum of nullity and rank of a transformation T equals the dimension of the domain U.
Similar Matrices
- Two nxn matrices A and B are similar if there exists an invertible matrix P such that B = P^-1AP, indicating they represent the same linear transformation under different bases.
Equivalence Relation of Similarity
- Every square matrix is similar to itself.
- If matrix A is similar to B, then B is similar to A.
- If A is similar to B and B is similar to C, then A is similar to C.
Diagonalizability
- A matrix A is diagonalizable if it can be expressed in a diagonal form through an appropriate change of basis.
Eigenvalues and Eigenvectors
- Eigenvalues are scalars associated with a linear transformation; eigenvectors are non-zero vectors that change only in scale when transformed, satisfying the equation Av = λv.
Characteristic Polynomial
- The characteristic polynomial of a matrix A is derived from det(A - λI), where I is the identity matrix and λ represents eigenvalues.
Finding Eigenvalues
- Eigenvalues can be found by solving the characteristic polynomial equation.
Trace of a Matrix
- The trace of a matrix A, denoted Tr(A), is the sum of the diagonal entries, providing insight into the sum of eigenvalues.
Characteristic Polynomial and Properties
- Two matrices have the same characteristic polynomial if they are similar; however, similarity is not guaranteed solely based on sharing a characteristic polynomial.
Link Between Trace, Determinant, and Eigenvalues
- The trace of a matrix equals the sum of its eigenvalues, while the determinant corresponds to the product of eigenvalues, linking matrix attributes to its spectral characteristics.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
This quiz covers key concepts related to linear transformations, including the kernel and image, null space, nullity, and rank. It also explores the rank-nullity theorem, which connects these properties in a comprehensive manner. Test your understanding of these fundamental topics in linear algebra.
