Proving Linear Independence and Basis for R3 Using Vector Determinants

In linear algebra, determining whether a set of vectors forms a basis for a vector space is a fundamental task. This article will guide you through the process of proving that the vectors (2 -1 4), (1 -1 2), and (3 1 -2) form a basis for (mathbb{R}^3). We will explore the concepts of linear independence and span, and use these to determine the validity of our vectors as a basis.

Step-by-Step Process

1. Understanding the Vectors and the Requirement for a Basis

For a set of vectors in (mathbb{R}^3) to form a basis, they must satisfy two conditions:

The vectors must be linearly independent. The vectors must span (mathbb{R}^3).

Since there are three vectors in (mathbb{R}^3), showing linear independence is sufficient to establish that they form a basis. In this case, we will focus on demonstrating linear independence using the determinant of a matrix formed by the vectors.

2. Forming the Matrix

We start by forming a matrix (A) with the vectors as rows:

[A begin{bmatrix} 2 -1 4 1 -1 2 3 1 -2 end{bmatrix}]

This matrix will help us calculate the determinant, which is crucial for determining linear independence.

3. Calculating the Determinant

Our goal is to calculate the determinant of the matrix (A). A non-zero determinant indicates that the vectors are linearly independent. We will use the rule of Sarrus to perform this calculation.

Expand the determinant along the first row: [text{det}A 2 cdot text{det}begin{bmatrix} -1 2 1 -2 end{bmatrix} - (-1) cdot text{det}begin{bmatrix} 1 2 3 -2 end{bmatrix} 4 cdot text{det}begin{bmatrix} 1 -1 3 1 end{bmatrix}] Calculate the 2x2 determinants: [text{det}begin{bmatrix} -1 2 1 -2 end{bmatrix} (-1)(-2) - (2)(1) 2 - 2 0] [text{det}begin{bmatrix} 1 2 3 -2 end{bmatrix} (1)(-2) - (2)(3) -2 - 6 -8] [text{det}begin{bmatrix} 1 -1 3 1 end{bmatrix} (1)(1) - (-1)(3) 1 3 4] Substitute these values back into the determinant formula: [text{det}A 2 cdot 0 - (-1) cdot (-8) 4 cdot 4 0 - 8 16 8]

4. Conclusion

Since the determinant (text{det}A 8) is non-zero, the vectors (2 -1 4), (1 -1 2), and (3 1 -2) are linearly independent. Therefore, they span (mathbb{R}^3) and form a basis for (mathbb{R}^3).

Final Result

The vectors (2 -1 4), (1 -1 2), and (3 1 -2) do indeed form a basis for (mathbb{R}^3).

Additional Information

Another method to ensure the vectors are not coplanar is by evaluating the determinant of the cross product of any two vectors and a third vector. The cross product of the vectors (mathbf{a}), (mathbf{b}), and (mathbf{c}) can be expressed as (mathbf{a} cdot (mathbf{b} times mathbf{c})). If the result is non-zero, the vectors are not coplanar and form a basis for (mathbb{R}^3).

Key Points to Remember

Linear Independence: Vectors are linearly independent if the determinant of the matrix formed by these vectors is non-zero. Determinant Calculation: Use the rule of Sarrus or cofactor expansion to compute the determinant of the matrix. Basis for R3: A set of vectors forms a basis for (mathbb{R}^3) if they are linearly independent.

By following these steps and understanding the underlying concepts, you can easily prove that a set of vectors forms a basis for (mathbb{R}^3) or any other vector space.

Keywords: linear independence, determinant, basis for R3