Staff Resources
Proposed Mathematics Standards for California Public Schools - Grades Eight through Twelve - Linear Algebra
The general goal in this discipline is that students learn the techniques of matrix manipulation so as to be able to solve systems of linear equations in any number of variables. Linear Algebra is most often combined with another subject, such as Trigonometry, Mathematical Analysis, or Pre-Calculus.
1. Students solve simultaneous linear equations in any number of variables using Gauss-Jordan elimination.
2. Students interpret linear systems as coefficient matrices and the Gauss-Jordan method as row operations on the coefficient matrix.
3. Students reduce rectangular matrices to row echelon form.
4. Students perform addition on matrices and vectors.
5. Students perform matrix multiplication, multiply vectors by matrices and by scalars.
6. Students demonstrate understanding that linear systems are either inconsistent (no solutions), have exactly one solution, or have infinitely many solutions.
7. Students demonstrate understanding of the geometric interpretation of vectors and vector addition (via parallelograms) for vectors in the plane and in three dimensional space.
8. Students interpret the solution sets of systems of equations geometrically. For example the solution set of a single linear equation in two variables is interpreted as a line in the plane, and the solution set of a two by two system is interpreted as the intersection of a pair of lines in the plane.
9. Students demonstrate understanding of the notion of the inverse to a square matrix, and apply it to solve systems of linear equations.
10. Students compute the determinants of 2 by 2 and 3 by 3 matrices,and are familiar with their geometric interpretations as area and volume of the parallelepipeds spanned by the images under the matrices of the standard basis vectors in 2-dimensional and 3-dimensional spaces.
11. Students know that a square matrix is invertible if, and only if,its determinant is non-zero. They can compute the inverse to 2 by 2 and 3 by 3 matrices using row reduction methods or Cramer's rule.
12. Students compute the scalar (dot) product of two vectors in n-dimensional space, and know that perpendicular vectors have zero dot product.
