Linear Algebra: Change of Basis in R^2 with Invertible Matrix R

To show that there exists an invertible 2x2 matrix R such that [v]_β = R*[v]_α, we need to demonstrate that the change of basis matrix from α to β exists and is invertible.

Let's denote the change of basis matrix from α to β as P, a 2x2 matrix. Our goal is to find a matrix P such that [v]_β = P*[v]_α for any vector v in R^2.

Since α=(u1, u2) and β=(w1, w2) are both bases of R^2, any vector v in R^2 can be expressed as a linear combination of the basis vectors:

v = a1u1 + a2u2 v can also be represented in terms of the β basis:

v = b1w1 + b2w2

To connect the coefficients a1, a2 and b1, b2, we can write the following equation:

b1w1 + b2w2 = a1u1 + a2u2

Because α and β are bases, their vectors are linearly independent. This allows us to equate the coefficients on both sides of the equation, resulting in:

b1 = a1 b2 = a2

Now, we can express the vector v in both α and β bases:

[v]_α = [a1, a2] [v]_β = [b1, b2] = [a1, a2]

From this, we can see that the change of basis matrix P is simply the identity matrix:

P = [1, 0] [0, 1]

Since P is the identity matrix, it is clearly invertible. Therefore, there exists an invertible 2x2 matrix R (which is equal to P) such that [v]_β = R*[v]_α.

To calculate R, we can directly use the coefficients of the basis vectors in both α and β. The columns of R are given by the coefficients of the basis vectors in the β basis, expressed in terms of the α basis. Therefore, we have:

R = [b1, b2] [b1, b2]

Substituting b1 = a1 and b2 = a2, we get:

R = [a1, a2] [a1, a2]

Thus, the matrix R can be calculated by simply taking the coefficients of the basis vectors in both α and β and arranging them in a 2x2 matrix.