Question

Consider the set of vectors \((-1,1,1),(1,-1,1),(1,1,-1)\) a. Use the Gram-Schmidt process to find an orthonormal basis for \(R^{3}\) using this set in the given order. b. What do you get if you do reverse the order of these vectors?

Short answer

The orthonormal basis of \(R^{3}\) of the vector set in the given order is \(\{(-1\/sqrt{3}, 1\/sqrt{3}, 1\/sqrt{3}), u_2, u_3\}\), where \(u_2\) and \(u_3\) are calculated as in steps 2 and 3. If the order is reversed, the resulting orthonormal basis will be different but still orthonormal.

Step-by-step solution

Find the first orthonormal vector

The first orthonormal vector \(u_1\) is simply gotten by normalizing the first vector. Normalize the first vector \((-1,1,1)\) by dividing it by its length. The length of \((-1,1,1)\) is \(sqrt {(-1)^2 + 1^2 + 1^2} = sqrt{3}\). So, \(u_1\) will be \((-1\/sqrt{3}, 1\/sqrt{3}, 1\/sqrt{3})\).

Find the second orthonormal vector

First, subtract the projection of the second vector onto the first orthonormal vector from the second vector. The projection vector is \(proj_{u_1} v_2 = (v_2 \cdot u_1) u_1\), where \(u_1\) is the first orthonormal vector from Step 1, and \(v_2 = (1,-1,1)\). Subtract this projection from \(v_2\) to get a vector that is orthogonal to \(u_1\). This orthogonal vector can then be normalized to get the second orthonormal vector \(u_2\).

Find the third orthonormal vector

In order to find the third orthonormal vector, subtract the projections of the third vector onto \(u_1\) and \(u_2\) from the third vector. Just like in Step 2, these projection vectors are calculated as \(proj_{u_1} v_3 = (v_3 \cdot u_1) u_1\) and \(proj_{u_2} v_3 = (v_3 \cdot u_2) u_2\), where \(v_3 = (1,1,-1)\). Subtract these projections from \(v_3\) to get a vector that is orthogonal to both \(u_1\) and \(u_2\). This orthogonal vector can then be normalized to get the third orthonormal vector \(u_3\).

Reverse the order for second part of the question

Repeat Steps 1 to 3 but now start with the last vector, then the second, and finally the first vector. This will yield a different orthonormal basis for \(R^{3}\).

Key concepts

Orthonormal Basis

Understanding the foundation of an orthonormal basis is crucial when dealing with the concept of vector spaces, particularly in \(R^{3}\). An orthonormal basis for a vector space consists of unit vectors that are perpendicular to each other. This unique property simplifies many mathematical operations, including computations in linear algebra and vector calculus.

When you construct an orthonormal basis using the Gram-Schmidt process, as seen in the exercise, you start with a set of vectors and convert them into vectors that are both orthogonal (at right angles to each other) and of unit length (normalized). The exercise guides you through finding such a basis for a set of vectors in \(R^{3}\). Once this process is applied, irrespective of the initial vectors' order, the result is a set of vectors that can be used to express any other vector in that space in a simplified manner. This is invaluable for a variety of applications, including solving systems of equations and performing computer graphics transformations.

Vector Normalization

Vector normalization is the process of converting any non-zero vector to a unit vector that points in the same direction. A unit vector is a vector with a length of one. Normalization is obtained by dividing each component of the vector by its total length or magnitude.

In our exercise example, the first step is to normalize the initial vector \( (-1,1,1) \) to create \( u_1 \) by dividing it by its length, calculated as the square root of the sum of its components squared, which is \( \sqrt{3} \). This step is vital in the Gram-Schmidt process as it sets the stage for creating subsequent orthogonal vectors. Normalized vectors are especially handy when you need to compare or project vectors since their length is uniform and equal to one, removing the influence of magnitude in such computations.

Vector Projection

Vector projection is a way to map one vector onto another, essentially 'projecting' the essence of one vector (the projection’s source) onto a line created by another (the projection’s target). It's like casting a shadow from one vector onto another, where the shadow represents the portion of the source vector that aligns with the target.

In the context of the Gram-Schmidt process detailed in the exercise, you calculate the vector projection of the second vector onto the first orthonormal vector to find a component that is parallel to the first vector. This projection is then subtracted from the original vector to find the orthogonal component. With each subsequent vector, you remove the components that are not orthogonal to the preceding orthonormal vectors. The mathematical formula to calculate the projection of vector \( \vec{v} \) onto vector \( \vec{u} \) is \( proj_{\vec{u}} \vec{v} = (\vec{v} \cdot \vec{u}) \vec{u} \). Once you've found the orthogonal components using projection, you can proceed to normalize them to add to the orthonormal basis set.