Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


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