https://r2.erweima.ai/mathbot/image/1727176126886j7uiorbf.jpg

The problem asks for a basis of the vector subspace \( W = \{(x, y, z) \in \mathbb{R}^3 : y = 2x\} \).

### Step-by-step solution:

1. The condition for vectors in \( W \) is \( y = 2x \), meaning any vector in \( W \) has the form:
   \[
   (x, y, z) = (x, 2x, z) = x(1, 2, 0) + z(0, 0, 1).
   \]
   Therefore, the vectors can be written as a linear combination of \( (1, 2, 0) \) and \( (0, 0, 1) \).

2. This implies that the vectors \( (1, 2, 0) \) and \( (0, 0, 1) \) span the subspace \( W \).

3. Since these two vectors are linearly independent (they are not scalar multiples of each other), they form a basis for \( W \).

Now, let's check the options:

- a) \( \{(1, 0, 0), (0, 2, 1)\} \): This is not a correct basis because the first vector does not satisfy the condition \( y = 2x \).
- b) \( \{(0, 2, 1)\} \): This is a single vector and cannot form a basis for a subspace of dimension 2.
- c) \( \{(1, 0, 0), (0, 1, 2)\} \): The vector \( (1, 0, 0) \) does not satisfy \( y = 2x \).
- d) \( \{(1, 2, 0)\} \): This vector satisfies \( y = 2x \), but a single vector is insufficient for a 2-dimensional subspace.

Thus, the correct answer should be **none of the given options** seem to form a correct basis for the subspace defined by \( y = 2x \). The basis should consist of \( (1, 2, 0) \) and \( (0, 0, 1) \), which is not explicitly listed.

Would you like further clarification or details on any step?

### Follow-up questions:
1. How do we determine the dimension of a subspace in general?
2. What is the difference between a span and a basis?
3. How can you check if a set of vectors is linearly independent?
4. Can you give an example of another vector subspace and its basis?
5. Why does a 2-dimensional subspace require exactly two vectors for its basis?

### Tip:
To find a basis, express the condition that defines the subspace in terms of parameters, and then find the independent vectors that span the subspace.

Una base del subespacio vectorial W = {(x, y, z) ∈ R^3 : y = 2x} es:

Learn how to find the basis of a vector subspace W defined by the condition y = 2x in R^3. This problem covers key concepts in linear algebra, such as linear independence, vector subspaces, and basis construction. Suitable for undergraduate students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation