Calculating an orthonormal basis for the subspace defined by the equation 2x + y - z = 0 involves understanding the concept of subspaces, orthogonal vectors, and the Gram-Schmidt process. This subspace represents a plane in three-dimensional space, and finding an orthonormal basis allows us to describe this plane using a set of mutually perpendicular unit vectors. This article will delve into the steps involved in calculating such a basis.
Understanding the Problem
The equation 2x + y - z = 0 defines a plane in three-dimensional space. Any vector (x, y, z) that satisfies this equation lies on this plane. To find an orthonormal basis for this subspace, we need to find two linearly independent vectors that lie on the plane and are orthogonal (perpendicular) to each other. We can then normalize these vectors to have unit length.
Finding a Basis for the Subspace
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Parameterize the Solution Space: We can express the solution space of the equation 2x + y - z = 0 in terms of parameters. Solving for z, we get: z = 2x + y
This means any vector (x, y, z) that satisfies the equation can be written as (x, y, 2x + y). We can rewrite this in terms of parameters: (x, y, 2x + y) = x(1, 0, 2) + y(0, 1, 1).
Therefore, the vectors (1, 0, 2) and (0, 1, 1) form a basis for the subspace.
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Check Linear Independence: These two vectors are linearly independent because neither can be expressed as a scalar multiple of the other. This means they span the entire subspace defined by the equation.
Applying the Gram-Schmidt Process
Now, we need to transform this basis into an orthonormal basis using the Gram-Schmidt process:
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Choose a First Vector: Let's take the first vector from the basis: v1 = (1, 0, 2).
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Project onto v1: We need to project the second vector v2 = (0, 1, 1) onto v1. The projection of v2 onto v1 is:
proj(v1) v2 = (v2 · v1 / ||v1||²) v1
Where:
- v2 · v1 is the dot product of v2 and v1
- ||v1||² is the squared magnitude of v1
Calculating these:
- v2 · v1 = (0)(1) + (1)(0) + (1)(2) = 2
- ||v1||² = (1)² + (0)² + (2)² = 5
Therefore, the projection of v2 onto v1 is:
proj(v1) v2 = (2/5)(1, 0, 2) = (2/5, 0, 4/5).
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Find the Orthogonal Vector: Subtract the projection from v2 to get a vector orthogonal to v1:
u2 = v2 - proj(v1) v2 = (0, 1, 1) - (2/5, 0, 4/5) = (-2/5, 1, 1/5).
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Normalize the Vectors: Now, we need to normalize both v1 and u2 to have unit length. The normalized vectors are:
- v1' = v1 / ||v1|| = (1, 0, 2) / √5 = (1/√5, 0, 2/√5).
- u2' = u2 / ||u2|| = (-2/5, 1, 1/5) / √(26/25) = (-2/√26, 5/√26, 1/√26).
Therefore, the orthonormal basis for the subspace defined by 2x + y - z = 0 is { (1/√5, 0, 2/√5), (-2/√26, 5/√26, 1/√26) }.
Verification of the Orthonormal Basis
We can verify that this basis is indeed orthonormal by checking the following:
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Orthogonality: The dot product of the two vectors should be zero:
(1/√5, 0, 2/√5) · (-2/√26, 5/√26, 1/√26) = (1/√5)(-2/√26) + (0)(5/√26) + (2/√5)(1/√26) = 0
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Unit Length: The magnitude of each vector should be 1:
||(1/√5, 0, 2/√5)|| = √((1/√5)² + (0)² + (2/√5)²) = 1
||(-2/√26, 5/√26, 1/√26)|| = √((-2/√26)² + (5/√26)² + (1/√26)²) = 1
Conclusion
In conclusion, by understanding the concept of subspaces, orthogonal vectors, and the Gram-Schmidt process, we successfully calculated an orthonormal basis for the subspace defined by the equation 2x + y - z = 0. This orthonormal basis provides a convenient and efficient way to represent the plane in three-dimensional space. The process involves parameterizing the solution space, finding a basis, applying the Gram-Schmidt process to orthogonalize and normalize the basis vectors, and finally verifying the resulting basis for orthonormality.
