Final answer:
The standard matrix of the linear transformation that first reflects points through the origin and then through the x1-axis is [[-1, 0], [0, 1]].
Step-by-step explanation:
The student is asking about finding the standard matrix of a linear transformation t: R² → R² that reflects points through the origin and then through the horizontal x1-axis. The process of finding the standard matrix for a given transformation involves understanding how the basis vectors are transformed.
First, reflecting a point through the origin is equivalent to multiplying each coordinate by -1, which can be represented by the matrix A = −I, where I is the identity matrix. Next, reflecting a point through the x1-axis means negating the second coordinate, which can be represented by the matrix B = [[1, 0], [0, -1]].
Since matrix multiplication is associative, the standard matrix of t can be found by multiplying matrix B by matrix A (B⋅A). The final standard matrix for t is M = [[-1, 0], [0, 1]].