Input vectors to find the orthonormal basis using the Gram-Schmidt process, with a step-by-step breakdown of the process.
Use this Gram-Schmidt calculator to orthonormalize the set of vectors step by step and find the orthonormal basis in an inner product space.
Follow these steps:
In linear algebra and numerical analysis, the Gram-Schmidt process transforms a set of independent vectors into an orthonormal basis. This basis spans the same subspace as the original vector set. The process constructs an orthogonal set of vectors, meaning their dot product is zero. In addition to that, each vector in the new basis has a unit length (magnitude of 1). This orthonormal basis is helpful for certain applications of algebra and beyond. It is widely used in various fields like computer science, physics, statistics, and machine learning, as well as in QR decomposition, etc.
The Gram-Schmidt formula helps to find the projection of one vector onto another.
\(\ u_i = v_i - \sum_{j=1}^{i-1} \frac{v_i \cdot u_j}{\|u_j\|^2} u_j \)
In algebra, orthogonal means the relationship between two vectors that are perpendicular to each other. Two vectors are orthogonal if the result of their dot product is zero.
For instance, if you have two following vectors:
\(\vec{u_1}, \vec{v_1}\) Then if \(\vec{u_1} \dot\ \vec{v_1}=0\)
They will be considered orthogonal. When it comes to orthogonal basis visualization, they can be easily visualized in two-dimensional Euclidean spaces. However, you can apply it to vectors having any number of dimensions.
For the sets of vectors having higher or comple dimensions, determining the orthogonal basis can be difficult and time taking. In such cases, tools like an orthogonal basis calculator, which uses the Gram-Schmidt process, are beneficial. The Gram-Schmidt process is an algorithm that transforms a set of linearly independent vectors into an orthogonal (or orthonormal) basis for the span of the vectors.
An orthonormal basis is a set of vectors within an inner product space (such as Euclidean space with the dot product) that spans the same space as the original vectors. These basis vectors are characterized by two key properties:
Using an orthonormal basis simplifies many vector space operations, including:
To facilitate the creation of such a basis, you can use an orthonormal basis calculator. It makes the calculations straightforward because the basis vectors are perpendicular to each other and have a unit length. This way, it forms a well-defined coordinate system for a vector space.
Orthonormalize the set of the vectors\(\ V_1 = \begin{bmatrix} 1 \\ 1 \\ \end{bmatrix} \ , V_2 = \begin{bmatrix} 2 \\ 1 \\ \end{bmatrix} \) , using the Gram-Schmidt process.
Solution:
Gram-Schmidt process,\(\vec{u_k} = \vec{v_k} - \sum_{j=1}^{k-1} \text{proj}_{\vec{u_j}} (\vec{v_k})\)
Where,
\(\ proj_{\vec{u_j}} (\vec{v_k}) = \frac{\vec{u_j} \cdot \vec{v_k}}{|\vec{u_j}|^2} \vec{u_j}\) is a vector projection.
The Normalized Vector is
\(\vec{e_k} = \frac{ \vec{u_k}}{|{\vec{u_k}}|}\)
Step 1:
\(\vec{u_1} = \vec{v_1} = \begin{bmatrix} 1\\ \ 1 \ \end{bmatrix}\)
\(\vec{e_1} = \frac{\vec{u_1}}{|\vec{u_1}|} = \frac{\begin{bmatrix} 1 \ 1 \end{bmatrix}}{\sqrt{1^2 + 1^2}} = \begin{bmatrix} \frac{1}{\sqrt{2}} \ \\\frac{1}{\sqrt{2}} \end{bmatrix} \approx \begin{bmatrix} 0.71 \\ 0.71 \\ \end{bmatrix}\)
Step #2: (Find Vector Projection)
\(\text{proj}_{\vec{u_1}} (\vec{v_2}) = \begin{bmatrix} 1.5 \\ 1.5 \\ \end{bmatrix}\)
Vector Subtraction:
\(\vec{u_2} = \vec{v_2} - \text{proj}_{\vec{u_1}} (\vec{v_2}) = \begin{bmatrix} 0.5 \\ -0.5 \end{bmatrix}\)
Calculate Unit Vector:
\(\vec{e_2} = \frac{ \vec{u_2}}{|{\vec{u_2}}|} \ = \ \begin{bmatrix} 0.71 \\ -0.71 \\ \end{bmatrix} \)
Result:
\(\begin{bmatrix} 0.71 \\ 0.71 \\ \end{bmatrix} \begin{bmatrix} 0.71 \\ -0.71 \\ \end{bmatrix} \)
Although the Gram-Schmidt process can be performed manually, specialized tools like a Gram Schmidt calculator can automate these calculations, especially for larger or more complex sets of vectors. They not only save time but also reduce the risk of human error.
The Gram-Schmidt process generates the orthonormal bases, and its applications are not limited to linear algebra. Let's take a look at some important areas where it plays a significant role:
1. Start with a set of linearly independent vectors:
\(\vec{u}_1,\ \vec{u}_2,\ \vec{u}_3,\ \ldots,\ \vec{u}_n\)
2. Set:
\(\vec{v}_1 = \vec{u}_1\)
Then normalize:
\(\vec{e}_1 = \frac{\vec{v}_1}{\|\vec{v}_1\|}\)
3. Compute the orthogonal vector for** \( \vec{u}_2 \):
\(\vec{v}_2 = \vec{u}_2 - \mathrm{proj}_{\vec{v}_1} \vec{u}_2 = \vec{u}_2 - \frac{\vec{u}_2 \cdot \vec{v}_1}{\vec{v}_1 \cdot \vec{v}_1} \vec{v}_1\)
Normalize it:
\(\vec{e}_2 = \frac{\vec{v}_2}{\|\vec{v}_2\|}\)
4. For \( \vec{u}_3 \), subtract the projections onto both \( \vec{v}_1 \) and \( \vec{v}_2 \):
\(\vec{v}_3 = \vec{u}_3 - \mathrm{proj}_{\vec{v}_1} \vec{u}_3 - \mathrm{proj}_{\vec{v}_2} \vec{u}_3\)
Normalize it:
\(\vec{e}_3 = \frac{\vec{v}_3}{\|\vec{v}_3\|}\)
5. For each \( \vec{u}_i \), subtracting projections onto all previous \( \vec{v}_j \) (for \( j < i \)), and normalize:
\(\vec{v}_i = \vec{u}_i - \sum_{j=1}^{i-1} \mathrm{proj}_{\vec{v}_j} \vec{u}_i = \vec{u}_i - \sum_{j=1}^{i-1} \left( \frac{\vec{u}_i \cdot \vec{v}_j}{\vec{v}_j \cdot \vec{v}_j} \vec{v}_j \right)\)
The following set forms the orthonormal basis:
\(\vec{e}_1, \vec{e}_2, \ldots, \vec{e}_n\)
To streamline this process, especially with numerous or high-dimensional vectors, try our Gram Schmidt calculator. It instantly delivers precise orthonormal bases with clear, step-by-step explanations.
Yes, you can apply the Gram-Schmidt Process to linearly dependent vectors but it will result in a zero vector during the process, making normalization impossible. In simple words, it finds linear dependency, but it can't produce the full basis if the vectors are not linearly dependent.
Orthogonal vectors are 90 degrees from each other. Orthonormal vectors are also orthogonal and have a vector length of 1.
The order of the vectors does not affect the resulting subspace spanned by the orthonormal basis. However, it can affect the decomposition pattern of vectors you end up with on that basis.
A Gram-Schmidt calculator implements the Gram-Schmidt orthogonalization process on a set of linearly independent vectors. Its primary function is to:
References:
From the source of Wikipedia: Gram–Schmidt process
From the source of math.hmc.edu: Gram–Schmidt Method
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