Math Problem Statement

Let p = 4x^2 - 5x + 7. Find the coordinate vector of p relative to the following basis for P2, p1 = 1, p2 = 1 + x, p3 = 1 + x + x^2.

Solution

To solve the problem, we need to find the coordinate vector of p(x)=4x25x+7p(x) = 4x^2 - 5x + 7 relative to the given basis {p1,p2,p3}={1,1+x,1+x+x2}\{p_1, p_2, p_3\} = \{1, 1 + x, 1 + x + x^2\}.

Step-by-Step Solution

  1. Express p(x)p(x) as a linear combination of the basis vectors: Let: p(x)=c1p1+c2p2+c3p3p(x) = c_1p_1 + c_2p_2 + c_3p_3 Substitute the basis vectors: 4x25x+7=c1(1)+c2(1+x)+c3(1+x+x2)4x^2 - 5x + 7 = c_1(1) + c_2(1 + x) + c_3(1 + x + x^2)

  2. Expand the right-hand side: 4x25x+7=c1+c2(1+x)+c3(1+x+x2)4x^2 - 5x + 7 = c_1 + c_2(1 + x) + c_3(1 + x + x^2) Distribute c2c_2 and c3c_3: 4x25x+7=c1+c2+c2x+c3+c3x+c3x24x^2 - 5x + 7 = c_1 + c_2 + c_2x + c_3 + c_3x + c_3x^2 Combine like terms: 4x25x+7=(c1+c2+c3)+(c2+c3)x+c3x24x^2 - 5x + 7 = (c_1 + c_2 + c_3) + (c_2 + c_3)x + c_3x^2

  3. Equate coefficients: Compare coefficients of 11, xx, and x2x^2 on both sides:

    • Constant term: c1+c2+c3=7c_1 + c_2 + c_3 = 7
    • Coefficient of xx: c2+c3=5c_2 + c_3 = -5
    • Coefficient of x2x^2: c3=4c_3 = 4
  4. Solve the system of equations:

    • From c3=4c_3 = 4, substitute into the other equations: c2+4=5    c2=9c_2 + 4 = -5 \implies c_2 = -9
    • Substitute c2=9c_2 = -9 and c3=4c_3 = 4 into c1+c2+c3=7c_1 + c_2 + c_3 = 7: c19+4=7    c1=12c_1 - 9 + 4 = 7 \implies c_1 = 12
  5. Coordinate vector: The coordinate vector of p(x)p(x) relative to the basis is: [p]=(c1,c2,c3)=(12,9,4)[p] = (c_1, c_2, c_3) = (12, -9, 4)

Final Answer:

The coordinate vector is: 12,9,4\boxed{12, -9, 4}

Would you like further clarification on any of these steps?

Here are 5 related questions to expand your understanding:

  1. How do you verify that the calculated coordinate vector correctly reconstructs p(x)p(x)?
  2. What properties of a basis make such a decomposition possible?
  3. How would the solution change if the basis vectors included higher powers of xx?
  4. Why is the system of equations derived by equating coefficients always consistent?
  5. How does this concept relate to solving systems of linear equations in matrix form?

Tip: Always confirm that the given set of vectors forms a basis before calculating coordinate vectors.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Coordinate Vectors
Polynomial Spaces

Formulas

p(x) = c1 * p1 + c2 * p2 + c3 * p3
Equating coefficients of like terms: c1 + c2 + c3 = constant, etc.

Theorems

Linear Independence
Basis and Dimension of Vector Spaces

Suitable Grade Level

Grades 11-12 or early undergraduate