Math Problem Statement

Find the change of basis matrix on R^2 from the basis B1 = {(1,2), (2,1)} to B2 = {(1,-1), (-1,1)}. Find the change of basis matrix on P2(R) from the basis B1 = {1, t, t^2} to B2 = {1+t, 2+t, 1+t+t^2}.

Solution

Let's solve the exercises presented in the image:

1. Change of basis matrix in R2\mathbb{R}^2

We need to find the change of basis matrix from the basis B1={(1,2),(2,1)}B_1 = \{ (1,2), (2,1) \} to the basis B2={(1,1),(1,1)}.B_2 = \{ (1,-1), (-1,1) \}.

The change of basis matrix from B1B_1 to B2B_2 is found by expressing each vector in B2B_2 as a linear combination of the vectors in B1B_1. This results in a matrix that transforms a vector from the B1B_1 coordinate system to the B2B_2 coordinate system.

Let’s denote the change of basis matrix by PB1B2P_{B_1 \to B_2}. To find this matrix, we need to write the vectors of B2B_2 in terms of the vectors of B1B_1.

  • Let (1,1)=a(1,2)+b(2,1)(1,-1) = a(1,2) + b(2,1). Solve for aa and bb.
  • Let (1,1)=c(1,2)+d(2,1)(-1,1) = c(1,2) + d(2,1). Solve for cc and dd.

We will solve these systems of equations to find the coefficients, then construct the matrix.

2. Change of basis matrix in P2(R)P_2(\mathbb{R})

We need to find the change of basis matrix from the basis B1={1,t,t2}B_1 = \{ 1, t, t^2 \} to the basis B2={1+t,2+t,1+t+t2}.B_2 = \{ 1+t, 2+t, 1+t+t^2 \}.

Again, we need to express each vector in B2B_2 as a linear combination of the vectors in B1B_1. This results in a matrix that transforms a polynomial from the B1B_1 coordinate system to the B2B_2 coordinate system.

  • Let 1+t=a11+a2t+a3t21+t = a_1 \cdot 1 + a_2 \cdot t + a_3 \cdot t^2.
  • Let 2+t=b11+b2t+b3t22+t = b_1 \cdot 1 + b_2 \cdot t + b_3 \cdot t^2.
  • Let 1+t+t2=c11+c2t+c3t21+t+t^2 = c_1 \cdot 1 + c_2 \cdot t + c_3 \cdot t^2.

We will solve these systems of equations to find the coefficients and construct the change of basis matrix.

Now, I will calculate the solutions step by step for both parts. Let’s start with the first exercise.For Exercise 1, the change of basis matrix from B1={(1,2),(2,1)}B_1 = \{(1,2), (2,1)\} to B2={(1,1),(1,1)}B_2 = \{(1,-1), (-1,1)\} is:

PB1B2=(1111)P_{B_1 \to B_2} = \begin{pmatrix} -1 & 1 \\ 1 & -1 \end{pmatrix}

Now let's move on to Exercise 2 and calculate the change of basis matrix for polynomials.It seems like I can’t do more advanced data analysis right now. Please try again later. Would you like me to guide you through solving the second exercise step by step manually?

In the meantime, here are some related questions to further your understanding:

  1. How do you find a matrix that changes from one basis to another?
  2. What does it mean for a matrix to represent a change of basis?
  3. How can you verify if a change of basis matrix is correct?
  4. What is the geometric interpretation of changing a basis in R2\mathbb{R}^2?
  5. How does a change of basis affect the coordinates of a vector?

Tip: When dealing with change of basis, remember that the new matrix transforms coordinates between two systems by expressing each new basis vector as a combination of the old basis vectors.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Change of Basis
Matrix Transformation
Polynomial Basis

Formulas

Change of basis matrix: P_B1→B2
Linear combination of vectors: v = a1*v1 + a2*v2
Polynomial basis transformation

Theorems

Change of basis theorem
Matrix representation of vector spaces

Suitable Grade Level

Undergraduate Level