Bases and coordinates in R²

Drag the circular handles on the two colored vectors in the right vector space to specify a basis.

The left vector space is the space of coordinates for the vector space to the right.

Drag the point in the left vector space to specify a coordinate c: the corresponding vector B(c) appears in the right vector space.

Drag the point in the right vector space to specify a vector v: the corresponding coordinate B^-1(v) appears in the left vector space.

If the black point is to escape the square region on either side, its position is "clamped." This is to make sure you can always see the point in both vector spaces at all times. The "preimage" of the boundary of the right vector space is represented as a parallelogram in the left, and its only purpose is to show the limits to the point's location.

If the vectors are linearly dependent (that is, if the basis matrix has a zero determinant), then dragging the black point is temporarily disabled.

The standard basis for the right vector space is shown as a light blue grid.

Guided tour

• With the (default) standard basis, move the point around. What is the relationship between the point and its coordinate?
• Move the green vector to (1,1) and the red vector to (-1,1). Move the point to (2,1) in the right vector space: check that its coordinates are (1.5,-0.5). Move the point to coordinate (1,1) in the left vector space: check that the corresponding vector is (0,2).
• Create a basis with determinant 1 and with determinant -1. What is the sign of the determinant measuring? Look at the spatial relationship between the green and red vectors in each case.
• Let's pretend that the horizontal axis in the right side represents the constant term of a polynomial, and the vertical axis the linear term, so (a,b) is a + bx. Move the green vector to 1 - x and the red vector to 1 + x. What is the coordinate of 2x? (Check that it is (-1,1).)

---

©2016 Kyle Miller