Question

Let T:R ² →P ₂ (R) be the linear map defined by T((a,b))=a+bx+(a−b)x ²

for all (a,b)∈R ² . Consider the bases B={(1,0),(0,1)} and C={1,x,x ² } of R ² and P ² (R) respectively.

Answer 1

The question pertains to demonstrating the invariance of the distance from a point to the origin under rotations of the coordinate system, which can be shown using the Pythagorean theorem where the sum of the squares of the coordinates remains constant.

Step-by-step explanation:

Understanding Linear Maps and Invariance Under Rotation

The question at hand involves a linear map T:R² → P₂(R) that transforms elements of the two-dimensional real number space R² into polynomials of degree at most 2. The given linear map is defined as T((a,b)) = a + bx + (a-b)x², which takes an ordered pair (a, b) and maps it to a quadratic polynomial. To show that the distance of a point to the origin is invariant under rotations, one must utilize a foundational concept in mathematics, the Pythagorean theorem, which in coordinate form is given by a² + b² = c².

This theorem can be applied to the coordinates of a point P(x, y) relative to the origin (0, 0), where the distance to the origin c = √(x² + y²). Regardless of how the coordinate system is rotated, the distance 'c' remains unchanged because the values of x² + y² remain the same in the rotated system, say x'² + y'², due to the rotational symmetry of circle equations.

The message is mixed with fragments of information related to different mathematical concepts such as complex numbers and exponential functions, but the crucial takeaway is the invariance of the distance to the origin under coordinate rotations, relying on the fundamental nature of the Pythagorean theorem.