Introduction

Basis vectors can:

• describe the different possible internal stress fields;
• describe mode shapes in vibrations, heat transfer, ...;
• simplify many problems;
• ...

Linear dependence

Vectors are linearly independent if

only when all coefficients are zero.

So you cannot express one base vector in terms of the rest.

Two vectors are linearly dependent if they are along the same line.

Three vectors are linearly dependent if they are in the same plane.

rank

The rank of a matrix is the number of independent rows, or columns.

Basis

A basis for an n-dimensional space is any set of n linearly independent vectors in the space.

Any arbitrary vector in the space can be uniquely expressed in terms of the basis:

Choosing the right basis for a problem can be very helpful.

Span:

What can be produced with a given set of basis vectors:

• One basis vector spans a line:

  

  is a 1D straight line through the origin.
• two basis vectors span a plane:

  

  is a 2D plane through the origin.
• three basis vectors span a 3D space:

  

  is a 3D space through the origin.

09/30/02 0:31:28