Math Problem Statement

これが一次独立ということを示してください。

Solution

To show that the vectors x+1x + 1, x21x^2 - 1, and x3+1x^3 + 1 are linearly independent, we need to verify that the only solution to the equation

a(x+1)+b(x21)+c(x3+1)=0a(x + 1) + b(x^2 - 1) + c(x^3 + 1) = 0

is a=0a = 0, b=0b = 0, and c=0c = 0.

  1. First, expand the given equation: a(x+1)+b(x21)+c(x3+1)=0a(x + 1) + b(x^2 - 1) + c(x^3 + 1) = 0 ax+a+bx2b+cx3+c=0ax + a + bx^2 - b + cx^3 + c = 0

  2. Combine like terms: cx3+bx2+ax+(ab+c)=0cx^3 + bx^2 + ax + (a - b + c) = 0

  3. For the above polynomial to be identically zero, the coefficients of each power of xx must be zero. This gives us the following system of equations:

c = 0 \quad \text{(coefficient of } x^3) \\ b = 0 \quad \text{(coefficient of } x^2) \\ a = 0 \quad \text{(coefficient of } x) \\ a - b + c = 0 \quad \text{(constant term)} \end{cases}$$ 4. From the first three equations, we get $$a = 0$$, $$b = 0$$, and $$c = 0$$. Substituting these into the fourth equation, we confirm it holds true: $$0 - 0 + 0 = 0$$ Since the only solution is $$a = 0$$, $$b = 0$$, and $$c = 0$$, the vectors $$x + 1$$, $$x^2 - 1$$, and $$x^3 + 1$$ are linearly independent. Would you like any additional details or have any questions? Here are some further questions you might be interested in: 1. How can we determine if a larger set of vectors is linearly independent? 2. What is the geometric interpretation of linear independence in three dimensions? 3. How do we find the basis of a vector space? 4. What is the rank of a matrix, and how is it related to linear independence? 5. Can you explain the concept of a null space and its significance in linear algebra? **Tip:** When checking for linear independence, it's useful to set up and solve the corresponding system of equations to ensure that the only solution is the trivial one.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Linear Independence

Formulas

a(x + 1) + b(x^2 - 1) + c(x^3 + 1) = 0

Theorems

Criteria for Linear Independence

Suitable Grade Level

Grades 10-12