The web-based module

**GridMaster** permits students to model a portion of

**R**^{2} and define vectors that coordinatize this vector space.

You will want to open the **GridMaster** web page in order to understand the following discussion. (Note: It will take a while to download JavaSketchpad initially, so please be patient.)

Open GridMaster in new window

You can use **GridMaster** to construct electronic multigrid paper. Start by moving the **green** and **blue** vectors -- click on a circle at the end of a vector and drag it to a new position. Then click on the appropriate button to display a grid, e.g., a green grid corresponding to the green vectors. You can resize the scale of the entire grid space by moving the point **e(1)** if you need more a larger or smaller range of values. In the upper left, you will see the coordinates of the point *P* (the red vector) with respect to the natural basis and then with respect to the bases [*P*]_{B} defined by the blue vectors and [*P*]_{G}defined by the green vectors.

Note: The "?" at the bottom right-hand corner of the workspace is a link to Key Curriculum Press and its About *JavaSketchpad* web page.

### Sample Exploratory Activity:

Find the vector *P* determined by the given coordinate vector [*P*](*u*,*v*) and the given basis {*u*,*v*}.

Given the vector *P* and a basis {*u*,*v*}, find the coordinate vector [*P*](*u*,*v*).

Use the information provided to determine the missing element.

- , , and . Find .
- , , and . Without attempting to coordinatize
*P* with respect to the standard basis, find a basis *G* that satisfies these conditions, if any exists.

This last question begins to motivate a need to construct a change-of-coordinates matrix in order to determine more readily if a solution exists. Although there are an infinite number of solutions, determining a single one using GridMaster requires careful attention to the interaction of vectors and how they impact the movement of the grid lines and values of the coordinatized vectors.

Next or page: **6.** Transformer2D Tool and Sample Activity

Next page: **5.** Discussion of Transformer2D Tool