Midterm I Review Sheet for Math 207, Fall 2011

Concepts to know

  1. The idea of a vector
  2. The geometric idea of vector addition, v+w
  3. The geometric idea of scalar multiplication, kv
  4. The geometric concept of the norm of a vector, ||v||
  5. The geometric concept of the dot product of two vectors, v·w
  6. The relationship of the dot product to the norm
  7. The projection of one vector onto another
  8. Geometric properties of the cross product of two vectors, v×w
  9. Algebraic properties of (combinations of) the dot product, scalar product and cross product
  10. The words orthogonal and normal
  11. An equation of a plane
  12. Equations of a line in space (parametric and vector forms)
  13. a linear equation, system of linear equaions; a solution of a linear equation or system; the solution set
  14. the coefficient and augmented matrices of a linear system
  15. elementary row operations; row equivalent matrices
  16. [reduced] row-echelon matrices
  17. [in]consisten linear systems
  18. the rank of a matrix; the Rank Theorem
  19. free variables in a solution of a linear system
  20. a homogeneous system; the trivial solution

Things to know how to do

  1. compute components of a vector connecting two points in the plane or in space
  2. compute components of v+w
  3. compute components of kv
  4. compute ||v|| from the components of v
  5. compute the dot product v·w with two different approaches:
    1. using the components
    2. using the geometric approach, in terms of the lengths of v and w and the angle between them
  6. compute the cross product v×w in two ways:
    1. using the complicated formula entirely with components
    2. using the geometric approach: it's direction (use the right hand rule!) and magnitude
  7. tell if two vectors are parallel
  8. tell if two vectors are orthogonal
  9. tell if a vector is orthogonal to a plane
  10. tell if a vector is parallel to a plane
  11. find the angle between two vectors
  12. find a vector along a line given the parametric or vector form of the line
  13. find a normal vector to a plane given its equation (or given three points on it)
  14. write down the equation of a plane given a normal and a point
  15. find a vector parallel ("along", "in the direction of") a line, given an equation for the line
  16. find a point on a line, given its equation
  17. find a point on a plane, given its equation
  18. write down the coefficient or augmented matrix corresponding to a linear system, and the system corresponding to a matrix
  19. apply particular EROs to matrix
  20. row-reduce a matrix to [R]REF
  21. perform back substitution to solve a linear system in RREF form
  22. do Gaussian Elimination
  23. do Gauss-Jordan Elimination
  24. determine when a system is [in]consistent and when it has infinitely many solutions