Math Problem Statement

Determine whether the vectors 1 - 2x, x^2, and x + x^2 of P_2 are linearly independent.

Solution

The question is asking whether the vectors 12x1 - 2x, x2x^2, and x+x2x + x^2 from the space P2P_2 (polynomials of degree at most 2) are linearly independent.

Step-by-step solution:

  1. Definition: A set of vectors (or polynomials) is linearly independent if the only solution to the equation: c1(12x)+c2x2+c3(x+x2)=0c_1(1 - 2x) + c_2x^2 + c_3(x + x^2) = 0 is c1=c2=c3=0c_1 = c_2 = c_3 = 0.

  2. Form the equation: Let's expand the linear combination: c1(12x)+c2x2+c3(x+x2)=0c_1(1 - 2x) + c_2x^2 + c_3(x + x^2) = 0 Expanding each term: c1(12x)=c12c1xc_1(1 - 2x) = c_1 - 2c_1x c3(x+x2)=c3x+c3x2c_3(x + x^2) = c_3x + c_3x^2 Putting them all together: (c12c1x)+c2x2+(c3x+c3x2)=0(c_1 - 2c_1x) + c_2x^2 + (c_3x + c_3x^2) = 0 Grouping terms by powers of xx:

    • Constant term: c1c_1
    • Linear term (xx): 2c1+c3-2c_1 + c_3
    • Quadratic term (x2x^2): c2+c3c_2 + c_3

    This gives the system of equations:

    c_1 &= 0 \quad \text{(from the constant term)} \\ -2c_1 + c_3 &= 0 \quad \text{(from the linear term)} \\ c_2 + c_3 &= 0 \quad \text{(from the quadratic term)} \end{aligned}$$
  3. Solve the system:

    • From c1=0c_1 = 0, substitute into the second equation: 2(0)+c3=0    c3=0-2(0) + c_3 = 0 \implies c_3 = 0
    • Substitute c3=0c_3 = 0 into the third equation: c2+0=0    c2=0c_2 + 0 = 0 \implies c_2 = 0
  4. Conclusion: The only solution to this system is c1=0c_1 = 0, c2=0c_2 = 0, and c3=0c_3 = 0. Therefore, the vectors are linearly independent.

Correct Answer:

B. They are linearly independent.

Would you like more details or have any questions?

Here are five related questions you could explore:

  1. What is the difference between linear independence and linear dependence?
  2. Can vectors in higher-dimensional polynomial spaces be linearly dependent?
  3. How does the method for determining linear independence change with more vectors?
  4. What is the role of the zero vector in determining linear independence?
  5. How does linear independence relate to the concept of a basis?

Tip: To test linear independence, always set up the linear combination and analyze the resulting system of equations.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Linear Independence
Vector Spaces
Polynomial Spaces

Formulas

Linear combination: c1(1 - 2x) + c2(x^2) + c3(x + x^2) = 0

Theorems

Theorem of Linear Independence in Vector Spaces

Suitable Grade Level

University Level (Undergraduate)