# Interactive Online Modules for Matrix Algebra and Linear Algebra

## Vector Spaces

#### Sample Investigations:

Linear Independence

#### Other Topics that Can Be Covered Using this Module:

• Linear Combination
• Linear Independence
• Span
• Spanning Set
• Basis
• Coordinates
• Vectors

#### Module Description

This module was developed using Mathematica, and it provides graphs of vector spaces in R3.

The module shows vectors and vector operations from a geometric perspective. It is divided into three sections, each providing different tools.

Under the section "For Collection of Linear Combinations," values of a, b and c are entered. The module uses the values of a, b and c to produce the geometric representation (as dots) of the collection of all linear combinations of the vectors, nk+ml+si+sj where k, l, i, and j are the vectors and the values of n range from –a to a; the values of m range from –b to b and the values of s range from –c to c.

Under the "Vectors" section, vectors are associated with the names k, l, i, j. Any vector can be entered including the stored vectors labeled a0-a14. Each vector is represented by a tick line segment with a color code. The module runs by Mathematica software. In Mathematica, vectors are entered using curly set notation. For instance, Mathematica recognizes {1, 2, 3} as a vector.

Under the "Construction of Single Linear Combination" section, you can assign values for the symbols d and e. These values produce thin lines which are parallel copies (with the same colors as the vectors represented) of scalar multiples of the vectors k, l. For instance, when the values d=2 and e=3 are entered, the program produces thin lines with the same color as the vectors k and l respectively, providing the length of the lines as twice the first vector k and three times the second vector l. This section also has an option to enter a w vector, which can be used to produce the picture of the vector coming from a linear combination of any vectors as well as k and l vectors.

Figures can be rotated by moving mouse, and they can be resized by using mouse and "Shift" button on your keyboard.