basis grids in the plane and their associated coordinate systems

This in class exercise practices visualizing linear changes of coordinates in the plane adapted to new bases. This gives a concrete picture to give context to the matrix manipulations we will do with applications to differential equations.

• bigblankgridprint0.pdf completed example done together in class step by step (handout in class)
• bigblankgridprint1.pdf exercise:  (to do)
• bigblankgridprint2.pdf exercise:  (to do)

source: basisgrids.mw, print exercises 1, 2 together (handout in class)

Example 0:  b₁ = <2,1>,  b₂ = <1,3> This is the template example used in the paper handout example redone with new points.

Example 1:  b₁ = <3,1>,   b₂ = <1,2>

Example 2: b₁ = <1,1>,  b₂ = <-2,1>

Axes.
For each case using a straight edge (piece of paper?) draw these arrows at the origin and label them by their symbols, and then replicate them tip to tail along the new coordinate axis lines through them, marking off the new tickmarks by small bullet circles at each new tip location. For example, starting at the origin: with <2,1>, move right 2, up 1, mark, repeat until hitting edge of old grid, then repeat in the opposite direction from the origin (left 2, down 1, etc). Then draw a line through them for the first axis with an arrowhead at the positive end, labeled by the new coordinate name y₁. Repeat for the second axis.

From old to new.
Locate the point with the old coordinates <x₁, x₂>. Draw a line from it to each new axis parallel to the other new axis to form a parallelogram. Again move tip to tail from the initial point as above until crossing the target axis. Read off the multiple of the basis vector along that axis. Repeat for the other axis. These multiples (chosen to be integers in these examples) are the new coordinates <y₁, y₂>.

From new to old.
Mark off the new coordinate multiples <y₁, y₂> of the two basis vectors along their axes. These new vectors are the sides of the parallelogram they form whose main diagonal is the position of the target point with coordinates  <x₁, x₂>. Starting at the tip of each of these two vectors, draw the second pair of parallelogram sides moving tip to tail as before parallel to the other new vector until they meet at the target point. Read off the old coordinates. This is a lot easier to understand watching bob in class do this stuff on Example 0.

Confirm by matrix multiplication.
Confirm your graphical read outs by the corresponding matrix multiplications as shown in Example 0. If you  have time, pick a point in the plane and project it onto the new axes for both examples and read off approximately what multiples of the basis vectors are obtained by this projection process (random points will not lead to integer values!)