Math 307 §B: Objectives for First Midterm Exam
State definitions of the following
- Row echelon form, reduced row echelon form.
- Pivot position, pivot column.
- Span of a set of vectors.
- Matrix-vector product.
- Linearly independent set of vectors.
- Linearly dependent set of vectors.
- Linear transformation.
- Onto mapping.
- One-to-one mapping.
- Matrix-matrix product.
Perform the following
- Perform vector addition and scalar multiplication of vectors, and interpret the results geometrically.
- Perform matrix-vector multiplication (A, x) -> Ax and interpret the result as
- a linear combination of the columns of A, and
- a linear transformation of the vector x.
- Use row reduction to solve a linear system.
- Determine whether a system Ax=b is consistent by the location of pivots in the augmented matrix [A ¦ b].
- Write the solution in parametric form.
- Express the solution of a homogeneous system as the span of a set of vectors.
- Conversely, express the span of a given set of vectors as the solution of a homogeneous system.
- Describe the solution set of a nonhomogeneous system Ax=b geometrically as the solution set of homogeneous system Ax=0 translated by a particular solution.
- Use the concepts of linear independence and linear dependence:
- Determine whether a given set of vectors is linearly independent.
- Use shortcuts to determine linear independence or dependence
of a set that
- consists of one vector only;
- consists of two vectors;
- contains the zero vector.
- For a linear transformation T:R^n -> R^m with standard matrix A=[a_1, ..., a_n]:
Relate the one to one property of T to
- The number of solutions of Ax=0;
- The number and location of pivots in A;
- Linear independence of the columns of A;
Relate the onto property of T to
- Consistency of Ax=b for all b in R^m;
- The number and location of pivots in A;
- Span{a_1, ..., a_n}.
- Perform matrix-matrix operations.
- Use the relation between the columns of a product AB and the columns of B.
- Interpret each elementary row operation as the result of left multiplication by an elementary matrix.
- Calculate algebraically with expressions involving matrices and their inverses.
- Determine whether an n by n matrix A is invertible, and, if it is, find its inverse.
- Use the shortcut for two by two matrices.
- State and use criteria for invertibility involving
- Row equivalence to I;
- Number of pivot positions in A;
- Number of solutions of Ax=0;
- Linear independence of columns of A;
- Consistency of Ax=b for each b in R^m;
- The span of the columns of A;
- The one to one property of the linear transformation x -> Ax;
- The onto property of the linear transformation x -> Ax;
- Existence of a left inverse of A;
- Existence of a right inverse of A.
- Express the inverse of a product of invertible matrices in terms of the inverses of the factors.
