Math 307 §D: Objectives for Second Midterm Exam
State definitions of the following.
- Subspace of a vector space.
- Null space, column space and row space of a matrix.
- Kernel and range of a linear transformation.
- Linear independence; linear dependence.
- Basis of a vector space or vector subspace.
- Coordinates of a vector relative to a basis.
- Dimension of a vector space.
- Rank of a matrix.
Do the following.
- Given a matrix reducible to echelon form without row interchanges, find an LU factorization of it.
- Compute the determinant of a matrix by cofactor expansion, exploiting row and column operations to simplify.
- Use the multiplicative property of determinants.
- Apply Cramer's Rule to represent the solution of a linear system.
- Represent elements of the inverse of a matrix in terms of cofactors.
- Interpret two by two and three by three determinants as signed areas and volumes.
- Relate the determinant of a matrix to invertibility, linear independence of columns or rows, etc.
- Determine whether a given set of vectors is a vector subspace.
- Find a basis for the null space of a matrix.
- Find a basis for the column space of a matrix.
- Find the kernel and range of a linear transformation.
- Find the coordinates of a vector relative to a basis.
- Apply the Rank Theorem.
- Relate the rank of a matrix to other properties in the invertible matrix theorem.
- Find the change-of-coordinates matrix from one basis to another.
