Problem 14 - Advanced Engineering Mathematics

Question

The given vectors span a subspace \(W\) of \(R^{3}\). Use the Gram-Schmidt orthogonalization process to construct an orthonormal basis for the subspace. $$ \mathbf{u}_{1}=\langle 1,2,3\rangle, \mathbf{u}_{2}=\langle 3,4,1\rangle $$

Step-by-Step Solution

Answer

An orthonormal basis is \( \{ \frac{1}{\sqrt{14}}\langle 1,2,3 \rangle, \frac{1}{\sqrt{3}}\langle 1,1,-1 \rangle \} \).

Step 1: Identify the vector basis for the subspace

We have two vectors given: \( \mathbf{u}_1 = \langle 1, 2, 3 \rangle \) and \( \mathbf{u}_2 = \langle 3, 4, 1 \rangle \). These vectors form a basis for the subspace \( W \) of \( \mathbb{R}^3 \). The goal is to transform these into an orthonormal basis using the Gram-Schmidt process.

Step 2: Apply Gram-Schmidt to find orthogonal vectors

First, set \( \mathbf{v}_1 = \mathbf{u}_1 = \langle 1, 2, 3 \rangle \).Next, compute \( \mathbf{v}_2 \) by making \( \mathbf{u}_2 \) orthogonal to \( \mathbf{v}_1 \):\[\mathbf{v}_2 = \mathbf{u}_2 - \frac{\mathbf{u}_2 \cdot \mathbf{v}_1}{\mathbf{v}_1 \cdot \mathbf{v}_1} \mathbf{v}_1\]Calculate the dot product \( \mathbf{u}_2 \cdot \mathbf{v}_1 = 3 \times 1 + 4 \times 2 + 1 \times 3 = 14 \).Calculate \( \mathbf{v}_1 \cdot \mathbf{v}_1 = 1^2 + 2^2 + 3^2 = 14 \).Thus, \( \mathbf{v}_2 = \langle 3, 4, 1 \rangle - \frac{14}{14} \langle 1, 2, 3 \rangle = \langle 3 - 1, 4 - 2, 1 - 3 \rangle = \langle 2, 2, -2 \rangle \).

Step 3: Find orthonormal vectors

Normalize the orthogonal vectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \):For \( \mathbf{v}_1 = \langle 1, 2, 3 \rangle \), find the norm:\[\| \mathbf{v}_1 \| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14}\]The unit vector \( \mathbf{e}_1 \) is:\[\mathbf{e}_1 = \frac{1}{\sqrt{14}} \langle 1, 2, 3 \rangle\]For \( \mathbf{v}_2 = \langle 2, 2, -2 \rangle \), find the norm:\[\| \mathbf{v}_2 \| = \sqrt{2^2 + 2^2 + (-2)^2} = \sqrt{12} = 2\sqrt{3}\]The unit vector \( \mathbf{e}_2 \) is:\[\mathbf{e}_2 = \frac{1}{2\sqrt{3}} \langle 2, 2, -2 \rangle = \frac{1}{\sqrt{3}} \langle 1, 1, -1 \rangle\]

Step 4: Verify orthonormal basis

The unit vectors \( \mathbf{e}_1 = \frac{1}{\sqrt{14}} \langle 1, 2, 3 \rangle \) and \( \mathbf{e}_2 = \frac{1}{\sqrt{3}} \langle 1, 1, -1 \rangle \) are orthonormal as both are unit vectors and orthogonal to each other. Therefore, \( \{ \mathbf{e}_1, \mathbf{e}_2 \} \) forms an orthonormal basis for the subspace \( W \).

Key Concepts

Orthonormal basis

An orthonormal basis for a vector space is a set of vectors that are both orthogonal to each other and have unit length. These vectors are particularly useful because they simplify calculations in many areas of mathematics, such as calculus and linear algebra.
To create an orthonormal basis, we typically start with a set of linearly independent vectors. However, these vectors are not necessarily orthogonal or have a unit norm. To solve this, we use the Gram-Schmidt orthogonalization process.

• First, we identify the initial set of basis vectors (the given vectors).
• Next, we apply the Gram-Schmidt process to make these vectors orthogonal.
• Finally, we normalize each orthogonal vector to unit length.

In our example, starting with vectors \( \mathbf{u}_1 = \langle 1, 2, 3 \rangle \) and \( \mathbf{u}_2 = \langle 3, 4, 1 \rangle \), the process led to the orthonormal basis \( \{ \mathbf{e}_1, \mathbf{e}_2 \} \). Each vector in this orthonormal basis satisfies the orthogonal and norm conditions, simplifying many operations like decomposition and projection.

Vectors in R3

Vectors in \( \mathbb{R}^3 \) are used to describe points or directions in three-dimensional space. These vectors have three components, commonly expressed as \( \langle x, y, z \rangle \).
In the process of solving problems involving vectors in \( \mathbb{R}^3 \), operations such as addition, subtraction, and scalar multiplication are frequently used. For example, in the Gram-Schmidt process, we calculate dot products and scale vectors:

• The dot product helps in determining orthogonality (perpendicularity).
• Adjusting vector length or direction involves scaling and addition/subtraction of vectors.

Being comfortable with these operations is crucial to effectively using vectors to solve real-world problems and mathematical tasks. In our case, the initial vectors \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) were transformed through such operations to obtain orthogonal vectors.

Subspace

A subspace in \( \mathbb{R}^3 \) refers to a vector space that is contained within a larger vector space. It retains the operations of the larger space, such as vector addition and scalar multiplication, but holds for a specific subset of vectors.
Subspaces have to satisfy certain conditions. First, they must include the zero vector. Secondly, the sum of any two vectors in the subspace must also be a vector in the subspace. Lastly, multiplying any vector in the subspace by a scalar must still yield a vector in the subspace.

• Spanned by given vectors: Any set of vectors can span a subspace if they are linearly independent.
• Maintain all properties of a vector space.

In the given exercise, the vectors \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) span a subspace \( W \) of \( \mathbb{R}^3 \), and through Gram-Schmidt, they are used to find an orthonormal basis for \( W \). This illustrates how particular sets of vectors define or help construct subspaces.