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Math 2700 Key Concepts

• systems of linear equations and writing them in matrix form
• augmented matrix
• Elementary Row Operations
• Existence and Uniqueness Questions
• Row Reduction, Echelon, and Reduced Echelon form
• Pivot positons
• Vector notation and equations
• Homogeneous and Inhomogeneous equations and systems
• Linear combinations and span
• Linear independence
• Connections between span, linear independence, existence questions, uniqueness questions, and pivots
• Linear transformations
 – Testing if a transformation is linear
 – writing a linear transformation as a matrix
 – basic geometric examples: rotation, dilation, shear
 – one-to-one (also called injective) and connections to linear independence and pivots
 – onto (also called surjective) and connections to span of the columns and pivots
 – invertible = one-to-one and onto (also called bijective)
 – kernel
 – range or image

• matrix addition, multiplication, and transpose
• How to invert a matrix
 – short-cut for 2 × 2
 – general procedure for 3 × 3 and larger

• Invertible Matrix Theorem
• null space
• column space
• determinants
 – calculating by expanding by cofactors
 – calculating by row operations
 – Cramer’s Rule
 – Connections between determinant of a linear transformation and volume
• Vector spaces and subspaces: definition and how to decide if a set is a vector space or subspace
• bases and dimension
• rank
• relationship between rank, dimension of null space, dimension of column space, and number of columns
• change of basis and coordinates
• Eigenvalues and eigenvectors
 – characteristic polynomial
 – imaginary eigenvalues
 – complex eigenvalues
 – Using eigenvalues and eigenvectors to help understand a linear transformation
• Connections between eigenvalues, eigenvectors, and differential equations as in the Romeo and Juliet examples
• Diagonalization and its relationship to eigenvalues and eigenvectors
• Dot products
 – angle between vectors
 – length of vectors
 – orthogonal
• orthogonal projections
• orthonormal bases and Gram-Schmidt orthogonalization
• least squares problems