Math 317 §B: Objectives for First Midterm Exam
State the definitions and use them in proof.
- vector, matrix
- linear combination of vectors
- dot product of vectors
- norm of a vector
- unit vector
- angle between vectors
- orthogonal vectors
- projection of a vector onto a nonzero vector
- matrix
- square matrix, diagonal matrix, identity matrix, triangular matrix
- transpose of a matrix
- augmented matrix of a linear system
- consistent linear system; inconsistent linear system
- row echelon form; reduced row echelon form
- homogeneous system
- equivalent linear systems; row equivalent matrices
- rank of a matrix
- row space of a matrix
- inverse of a matrix
- powers of a square matrix; inverse powers of a nonsingular matrix
- singular matrix, nonsingular matrix
State the Theorems and use them in proof.
- The Cauchy-Schwarz inequality (including the condition for equality).
- Triangle Inequality (including the condition for equality).
- Decomposing a vector into components parallel to and orthogonal to a nonzero vector.
- Row equivalence of every matrix to a unique reduced row echelon form.
- The relation between rank and number of solutions of a homogeneous system.
Algorithms: Do the following.
- Perform vector addition and scalar multiplication of vectors, and interpret the results geometrically.
- Compute dot products of vectors, and interpret the result geometrically.
- Express the norm of a vector in terms of the dot product.
- Compute the projection of a vector on a nonzero vector; given vectors a,b with b not zero, write a as the sum of a vector parallel to b and a vector orthogonal to b.
- Perform matrix-vector multiplication (A, x) -> Ax and interpret the result as
- a linear combination of the columns of A, and
- a vector of dot products of x with the rows of A.
- Find the transpose of a matrix (the conjugate transpose of a complex matrix). Identify symmetric and skew-symmetric real matrices. Identify Hermitian and skew-Hermitian complex matrices.
- Perform matrix-matrix operations.
- Use the relation between the columns of a product AB and the columns of B.
- Identify commuting matrices.
- Calculate algebraically with expressions involving matrices and their inverses.
- Interpret each elementary row operation as the result of left multiplication by an elementary matrix.
- 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 vector parametric form.
- 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.
- Express the solution of Ax=b in terms of b and A^(-1).
- 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.
- Express the inverse of a product of invertible matrices in terms of the inverses of the factors.
- State and use criteria for invertibility involving
- Row equivalence to I;
- Number of pivot positions in A;
- Number of solutions of Ax=0;
- Consistency of Ax=b for each b in R^n;
- Existence of a left inverse of A;
- Existence of a right inverse of A;
- Expression of A as a product of elementary matrices.
