Linear Algebra II

September 1, 2014

Lecture 1

Filed under: 2014 Fall — Y.K. Lau @ 3:54 PM

There are many ways to introduce linear algebra. Quite often one starts with system of linear equations, for example,

\displaystyle  \left\{\begin{array}{lcr} 2x+y&=& 8\\ x+3y & =& 9. \end{array} \right.

Everybody (in our course) knows how to solve it — using the method of elimination.

Algebraically, there are two important things stemming from this simple example:

  • (A) Matrix notation — a shorthand for the system.
  • (B) The method of Gaussian elimination.

The method in (B) involves three types of operations:

  • (I) Interchange two rows.
  • (II) Multiply one row by a nonzero number.
  • (III) Add a multiple of one row to another row.

These operations are called elementary row operations of type (I), (II), (III) resp.. All systems of linear equations can be solved by these three elementary row operations with a suitable algorithm — Gaussian Algorithm (also called the method of Gaussian elimination). [You are supposed to know it. If you don’t or you forgot, please study Chapter 1, Sections 1.1-1.2 (p. 1-17), of the textbook and do some exercises therein.]

Geometrically, we can also give an interpretation for systems of linear equations. Firstly let us link up geometric vectors with matrices: A geometric vector is an “arrow” which carries two pieces of information: magnitude (its length) and direction.

Geometric vectors can be added up by joining the head and tail. To describe this addition algebraically we represent a geometric vector (in a plane) by a {2\times 1} matrix, say, {\begin{pmatrix} x_1\\ y_1\end{pmatrix}}. Suppose the geometric vectors {\vec{a}} and {\vec{b}} are represented by

\displaystyle  \vec{a} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}, \quad \vec{b} = \begin{pmatrix} 1 \\ 3\end{pmatrix}, \quad \vec{c} = \begin{pmatrix} 8\\ 9\end{pmatrix}.

Then {\vec{a}+\vec{b}=\begin{pmatrix} 3\\ 4\end{pmatrix}}.

The scalar multiplication of a geometric vector can also be represented by a matrix operation, for example:

{2\vec{a}=\begin{pmatrix} 4\\ 2\end{pmatrix} = 2 \begin{pmatrix} 2 \\ 1\end{pmatrix}},     and    {3\vec{a}+2\vec{b}= \vec{c}}.

Now, we can give a geometric interpretation of the system { \left\{\begin{array}{lcr} 2x+y&=& 8\\ x+3y & =& 9. \end{array}\right. } Observe that the system can be expressed as

\displaystyle  x\begin{pmatrix} 2 \\ 1\end{pmatrix} + y \begin{pmatrix} 1 \\ 3\end{pmatrix} = \begin{pmatrix} 8\\ 9\end{pmatrix}     i.e.   {x\vec{a}+y \vec{b}=\vec{c}}.

Thus, to solve the system, it is equivalent to find {x,y} such that the vector sum {x\vec{a}+y\vec{b}} equals {\vec{c}}. We have observed that {3\vec{a}+2\vec{b}= \vec{c}}; consequently {x=3,y=2} is a solution for the system.

The concept of vectors can be extended to other objects, i.e. not merely matrices, so that we may apply the methodologies in matrices to these objects. (Mathematics is powerful as it can be applied to various objects!) Hence, what we did in today lecture is: Define Vector Spaces; Look at Examples & a Non-example.

 

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