Relationship Between Coordinate Vectors in Different Bases of R^2

To show that there exists a 2x2 matrix R such that [v]_β = R*[v]_α, we need to demonstrate that the coordinate vectors [v]_β and [v]_α are connected by a linear transformation.

Let v be a vector in R^2. Since α and β are bases of R^2, we can express v as a linear combination of the basis vectors:

v = au1 + bu2 (1)

v = cw1 + dw2 (2)

where a, b, c, and d are scalars.

Now, let's express [v]_α and [v]_β in terms of the coefficients a, b, c, and d:

[v]_α = [a; b] (3)

[v]_β = [c; d] (4)

From equations (1) and (2), we can rewrite [v]_α and [v]_β in terms of the basis vectors:

[v]_α = [a; b] = a*[u1]_α + b*[u2]_α (5)

[v]_β = [c; d] = c*[w1]_β + d*[w2]_β (6)

Since α and β are bases, we can express the basis vectors [u1]_α, [u2]_α, [w1]_β, and [w2]_β in terms of the standard basis vectors [e1] and [e2]:

[u1]_α = [x; y] (7)

[u2]_α = [z; w] (8)

[w1]_β = [s; t] (9)

[w2]_β = [p; q] (10)

Substituting equations (7)-(10) into equations (5) and (6), we get:

[a; b] = a*[x; y] + b*[z; w] (11)

[c; d] = c*[s; t] + d*[p; q] (12)

Now, let's combine equations (11) and (12) into a single matrix equation:

[a; b] = [x; y; z; w] * [a; b] = [s; t; p; q] * [c; d] (13)

where [x; y; z; w] and [s; t; p; q] are the columns of the 2x4 matrix R:

R = [x s; y t; z p; w q] (14)

Finally, we can rewrite equation (13) as:

[a; b] = R * [c; d] (15)

which shows that [v]_β = R * [v]_α, as desired.

Therefore, there exists a 2x2 matrix R such that [v]_β = R * [v]_α.