Program, Semester Week 8#
Matrix Algebra and Determinants#
This week concerns vectors and matrices. A matrix was introduced last week as a convenient notation form when solving systems of linear equations - this week we will se that a matrix can be considered a mathematical object in its own right. We will in this regard study how vectors and matrices can be multiplied together and how arithmetic operations and properties can be defined on such objects. Also, the important concept of linear dependence vs. linear independence will be treated, a topic that will become immensely important in the following weeks.
This week will also introduce the concept of determinants, a property of square matrices that often is useful. As we will see, the determinant can be used to investigate easily whether vectors are linearly independent or not.
Key Terms#
Vectors, scalars, and matrices \(\mathbf A\) as mathematical objects. Matrix arithmetic, including matrix multiplication, matrix-vector multiplication and scalar multiplication. Linear independency and linear dependency. Square matrix. The identity matrix \(\mathbf I_n\). Inverse matrix \(\mathbf A^{-1}\). Determinant \(\mathrm{det}(\mathbf A)\). The expansion theorem or Laplace expansion.
SymPy Demos#
From this week on we will be using the Python package SymPy as our CAS tool along with Jupyter Notebook. SymPy demos can be found via the menu on a weekly basis and we recommend that you read them through to train your use of SymPy alongside your mathematical learning. For this week we have the following two demos, the first of which shows the fundamentals of SymPy and Jupyter Notebook:
In case of installation issues, remember that the Python support is located in b.302, ground floor, every day for drop-in.
Preparation and Syllabus#
This week covers Sections 7.1, 7.2, and 7.3 until and including Equation (7-10) from Chapter [07 - Vectors and matrices] as well as Sections 8.1, 8.2 until and including Example 8.2.1, and 8.3 until and including Corollary 8.3.6 from Chapter [08 - Determinants].
Exercises#
Exercises for Long Day.
Exercises for Short Day.