math111_logo Change of Basis

1. Matrix for change of coordinates

Let B = {b1, b2, b3} and C = {c1, c2, c3} be two bases of a vector space V. Then the vectors in B may be written as linear combinations of vectors in C.

b1 = a11c1 + a12c2 + a13c3
b2 = a21c1 + a22c2 + a23c3
b3 = a31c1 + a32c2 + a33c3

The matrix

[B]C = [ [b1]C [b2]C [b3]C ] = [ a11 a21 a31 ],
a12 a22 a32
a13 a23 a33

with a very suggestive notation, is the "coordinates of B with respect to C". The next result suggests that [B]C is the matrix for changing the coordinates in B to the coordinates in C.

The coordinates of vV in the bases B and C are related by

[v]C = [B]C[v]B.

Proof Without loss of generality, we may assume V = R3. We use the bases vectors as columns and construct invertible matrices [B] = [b1 b2 b3], [C] = [c1 c2 c3]. Then

[B] = [a11c1 + a12c2 + a13c3 a21c1 + a22c2 + a23c3 a31c1 + a32c2 + a33c3]

= [c1 c2 c3] [ a11 a21 a31 ] = [C][B]C.
a12 a22 a32
a13 a23 a33

Let the coordinates of vV to be [v]B = (x1, x2, x3) and [v]C = (y1, y2, y3). Then

v = x1b1 + x2b2 + x3b3 = [B][v]B = [C][B]C[v]B.

Compared with

v = y1c1 + y2c2 + y3c3 = [C][v]C,

we have [B]C[v]B = [v]C.

We list some properties of the matrix for changing the coordinates.

transitivity
[C]D [B]C = [B]D
inverse
[B]C-1 = [C]B
identity
[B]B = I

The transitivity property means that changing the B-coordinates to the C-coordinates and then further changing to the D-coordinates is the same as changing the B-coordinates to the D-coordinates. The identity property means that the change from the B-coordinates to the B-coordinates is no change at all. The inverse follows from the transitivity property (take D = B) and the identity property.


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