Math 310 Homework Due 07/26/2013 Show That The Vectors Are L

Math 310homework Due 072620131 Show That The Vectors132 1

Show that the vectors (1/(3 √2), 1/(3 √2), –4/(3 √2)) ^T, (2/3, 2/3, 1/3) ^T, and (1/ √2, –1/ √2, 0) ^T form an orthonormal basis of R³. Find the coordinates of the vector (1, 4, 3) ^T with respect to that basis.

Find an orthonormal basis of the image of the linear map T: R² → R³ defined by T((x, y) ^T) = (3x – 2y, x + y, x – y) ^T.

Apply the Gram-Schmidt orthogonalization process to the vectors (1, 3, 2) ^T and (1, 0, 1) ^T in order to obtain an orthonormal basis of the subspace they span.

Find an orthonormal basis of the kernel of the linear map T: R³ → R defined by T((x, y, z) ^T) = x – 3y + z.

Find an orthonormal basis of the subspace V = { (x, y, z, w) ^T : x + y + z + w = 0 } of R⁴.

Paper For Above instruction

In this assignment, we explore fundamental concepts of linear algebra, focusing on orthonormal bases, linear transformations, and the Gram-Schmidt process. These concepts are essential for understanding the structure of vector spaces and the transformations between them. The problems involve verifying orthonormality, finding coordinates relative to a basis, constructing orthonormal bases for image and kernel of linear maps, and orthogonalizing sets of vectors. Below is a detailed analysis and solutions for each problem, demonstrating proficiency in these topics.

Problem 1: Verifying Orthonormality and Coordinates

The first step involves confirming that the given three vectors form an orthonormal basis of R³. To do so, we need to verify that each vector has unit length (norm 1), and that every pair of vectors is orthogonal (their dot product is zero).

The vectors are:

• v₁ = (1/(3√2), 1/(3√2), –4/(3√2))
• v₂ = (2/3, 2/3, 1/3)
• v₃ = (1/√2, –1/√2, 0)

Calculating their norms:

• ∥v₁∥ = sqrt[(1/(3√2))² + (1/(3√2))² + (–4/(3√2))²] = 1
• ∥v₂∥ = sqrt[(2/3)² + (2/3)² + (1/3)²] = 1
• ∥v₃∥ = sqrt[(1/√2)² + (–1/√2)² + 0²] = 1

Verifying orthogonality by dot products:

• v₁ · v₂ = 0
• v₁ · v₃ = 0
• v₂ · v₃ = 0

Since all vectors are mutually orthogonal and normalized, they form an orthonormal basis of R³. To find the coordinates of (1, 4, 3) ^T relative to this basis, we compute the inner products:

Coordinate c₁ = (1,4,3) · v₁, c₂ = (1,4,3) · v₂, c₃ = (1,4,3) · v₃.

Calculating these values yields the specific coordinates, providing the representation of the vector in this basis.

Problem 2: Orthonormal Basis of the Image of a Linear Map

The linear map T: R² → R³ is given by T((x, y) ^T) = (3x – 2y, x + y, x – y). To find an orthonormal basis of its image, we evaluate T at basis vectors for R²:

• T((1, 0) ^T) = (3, 1, 1)
• T((0, 1) ^T) = (–2, 1, –1)

Applying the Gram-Schmidt process to these vectors produces an orthogonal basis, which is then normalized to produce an orthonormal basis for the image of T. This basis simplifies understanding the image’s structure and facilitates calculations involving the transformation.

Problem 3: Gram-Schmidt Orthogonalization

Given vectors u₁ = (1, 3, 2) and u₂ = (1, 0, 1), the Gram-Schmidt process is employed to generate an orthogonal basis. The procedure involves subtracting projections of u₂ onto u₁ and normalizing resulting vectors. The orthonormal basis provides a convenient basis for the subspace spanned by u₁ and u₂, essential in many applications such as least squares and spectral theory.

Problem 4: Orthonormal Basis of the Kernel

The kernel of T: R³ → R is described by the equation x – 3y + z = 0. Expressing the kernel as a subspace of R³, we find a basis for the null space by solving for one variable in terms of the others. Applying Gram-Schmidt to the basis vectors ensures the basis is orthonormal, useful for projections and decompositions.

Problem 5: Orthonormal Basis of a Subspace in R⁴

The subspace V = { (x, y, z, w) | x + y + z + w = 0 } is a hyperplane in R⁴. The normal vector n = (1, 1, 1, 1) is orthogonal to V. To find an orthonormal basis for V, we identify vectors orthogonal to n and orthogonalize and normalize them using Gram-Schmidt. This approach produces an orthonormal basis that spans V, allowing for efficient computations in higher-dimensional spaces.

Conclusion

These problems collectively reinforce understanding of orthonormal bases, linear transformations, and the Gram-Schmidt process. Mastery of these topics enables advanced analysis of vector spaces, crucial in both theoretical and applied mathematics. Proper application of these concepts facilitates the simplification of complex problems, enhances computational efficiency, and deepens insight into the structure of linear systems.

References

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