Question

Let β denote the standard basis for R³ and set α= {(1 3 0 ), (0 1 -1), (-1 1 1)} (a) Determine the change of basis matrix P that changes from the α basis to the β basis. (b) Using the change of basis matrix P found in part (a), find the coordinates of a vector v in the β basis given its coordinates in the α basis. You can proceed with determining the change of basis matrix and finding the coordinates of a vector in the specified bases.

Answer 1

To find the change of basis matrix P from the α basis to the β basis, solve the equation α = Pβ. To find the coordinates in the β basis given the coordinates in the α basis, use the equation vβ = Pvα.

Step-by-step explanation:

To find the change of basis matrix P that changes from the α basis to the β basis, we need to express each vector in the α basis as a linear combination of the vectors in the β basis. This can be done by solving the equation α = Pβ, where α and β are matrices representing the vectors in their respective bases, and P is the change of basis matrix we are trying to find. Rearranging the equation, we get P = αβ⁻¹. Using this equation, we can calculate P by first finding the inverse of the β matrix and then multiplying it with the α matrix.

To find the coordinates of a vector v in the β basis given its coordinates in the α basis, we can use the equation vβ = Pvα, where vβ represents the coordinates of v in the β basis, Pvα represents the change of basis matrix P multiplied by vα (the coordinates of v in the α basis), and vα represents the coordinates of v in the α basis. By multiplying the change of basis matrix P with the coordinates of v in the α basis, we can obtain the coordinates of v in the β basis.