The Gram-Schmidt Process

Professor Dave Explains
28 Jun 201910:06
EducationalLearning
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TLDRProfessor Dave's tutorial delves into the Gram-Schmidt Process, a mathematical method used to transform any set of basis vectors in a vector space into an orthogonal, and subsequently, an orthonormal basis. Starting with vectors that are not necessarily orthogonal, Dave explains step-by-step how to apply the Gram-Schmidt Process, beginning with the initial vector and systematically adjusting subsequent vectors to ensure orthogonality and normalization. Through a detailed example involving basis vectors in R3, the tutorial breaks down the process into comprehensible steps, showcasing how to achieve an orthonormal basis. This technique is essential for simplifying calculations in various vector spaces, including those involving functions, by providing a method to compute inner products uniquely.

Takeaways
  • 😀 The Gram-Schmidt process generates an orthogonal basis from any set of vectors.
  • 😉 It involves subtracting projections of earlier vectors from later vectors.
  • 🤓 Each new vector has terms subtracted for all previous vectors.
  • 🧐 The process uses vector dot products and vector lengths.
  • 👍 First vector is just the original first basis vector.
  • 👏 Later vectors subtract out overlap with previous vectors.
  • 💡 Subtractions use projection fractions based on dot products.
  • 📝 The new orthogonal vectors can be normalized to be orthonormal.
  • 🔍 An example shows the full process for a 3D basis.
  • 🤓 Understanding Gram-Schmidt helps with computations using orthogonal bases.
Q & A

  • What is the Gram-Schmidt process used for?

    -The Gram-Schmidt process is used to generate an orthogonal or orthonormal set of basis vectors from any given set of basis vectors.

  • How do you construct the orthogonal vectors u1, u2, etc. in the Gram-Schmidt process?

    -u1 is set equal to v1. u2 is set equal to v2 minus the projection of v2 onto u1. u3 is set equal to v3 minus the projections of v3 onto u1 and u2. The process continues similarly for higher order vectors.

  • Why is the Gram-Schmidt process useful?

    -The Gram-Schmidt process is useful because orthogonal and orthonormal bases have many nice mathematical properties that make calculations involving them easier and more efficient.

  • What is the difference between an orthogonal and orthonormal set of vectors?

    -An orthogonal set of vectors have dot products equal to zero with each other. An orthonormal set is orthogonal and also normalized to unit length.

  • What is the dot product used for in the Gram-Schmidt process?

    -The dot product is used to compute the projections of the original vectors onto the orthogonal basis vectors. These projections are subtracted to make the vectors orthogonal.

  • How do you normalize the orthogonal vectors to make them orthonormal?

    -Once the orthogonal vectors are constructed, they can be normalized by dividing each vector by its own length to make the lengths unity.

  • Does the Gram-Schmidt process work for any set of basis vectors?

    -Yes, the Gram-Schmidt process can take any linearly independent set of vectors and construct an orthogonal set from them.

  • Why is summation notation used for the higher order vectors?

    -The summation notation compactly represents having to subtract the projections onto all previously constructed orthogonal vectors, which increases for each new vector.

  • How does the Gram-Schmidt process change for different vector spaces?

    -The process stays the same, but the definition of the inner product changes. For example, with function spaces the inner product becomes an integral.

  • What is an example application of using an orthonormal basis?

    -Orthonormal bases are very useful in quantum mechanics for representing quantum states and operators.

Outlines
00:00
😀 Introducing the Gram-Schmidt Process

Paragraph 1 introduces the Gram-Schmidt process as a method for generating an orthonormal set of vectors from any given set of basis vectors. It explains that we start with basis vectors v1 to vn and derive new orthogonal vectors u1 to un using this process. The process involves subtracting off any components of the current v vector that are not orthogonal to previously generated u vectors.

05:06
📝 Step-by-Step Explanation and Example

Paragraph 2 provides a step-by-step explanation of the Gram-Schmidt process using mathematical notation and a concrete example with 3-dimensional basis vectors. It shows how to compute the u vectors by subtracting dot product terms involving previously generated u's from each v vector. It then works through a full example deriving an orthogonal set from the given basis.

Mindmap
Keywords
💡Gram-Schmidt Process
The Gram-Schmidt Process is a method used to convert a set of vectors into an orthogonal (or orthonormal) set of vectors. This is especially useful when starting with vectors that are not orthogonal. In the context of the video, the process is explained as a systematic way to achieve an orthonormal basis from any given set of basis vectors, thus making it a foundational concept for understanding linear algebra and vector spaces. The explanation provided through a step-by-step procedure illustrates how each vector in the original set is adjusted to remove components that are not orthogonal to the previously established vectors in the new set.
💡Orthogonal Vectors
Orthogonal vectors are vectors that meet at a right angle (90 degrees) to each other. In the video, the concept of orthogonal vectors is a precursor to introducing the Gram-Schmidt Process. Understanding orthogonal vectors is crucial because the Gram-Schmidt Process relies on this property to create a set of vectors where each is orthogonal to the others, ensuring that the vectors do not share any direction.
💡Orthonormal Vectors
Orthonormal vectors are orthogonal vectors that have been normalized so that each vector's length is 1. The video emphasizes the transition from orthogonal to orthonormal vectors by dividing each orthogonal vector by its length. This concept is key in simplifying computations in vector spaces, as orthonormal vectors maintain their perpendicular relationships while being easy to work with due to their unit length.
💡Vector Space
A vector space is a collection of vectors that can be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers. The video discusses creating an orthonormal basis for a vector space, highlighting the importance of understanding vector spaces for applying the Gram-Schmidt Process and the broader context of linear algebra.
💡Basis of a Vector Space
A basis of a vector space is a set of linearly independent vectors that span the entire space. In other words, any vector in the space can be expressed as a linear combination of the basis vectors. The video's focus on converting any set of basis vectors into an orthonormal set using the Gram-Schmidt Process underscores the basis's role in understanding and working within vector spaces.
💡Inner Product
The inner product (or dot product, in the context of real vectors) is a way of multiplying two vectors to yield a scalar. The video uses the inner product extensively in the Gram-Schmidt Process to adjust vectors and ensure orthogonality by projecting one vector onto another. The calculation of the inner product is crucial for determining the components of vectors that are parallel and must be subtracted to achieve orthogonality.
💡Normalization
Normalization is the process of adjusting the length of a vector to 1 without changing its direction. In the video, normalization is the final step in converting orthogonal vectors into orthonormal vectors. This is done by dividing each vector by its magnitude (length), which simplifies many operations in linear algebra by working with vectors of uniform length.
💡Vector Magnitude
Vector magnitude (or length) is the measure of the length of a vector. In the video, the magnitude is calculated using the square root of the dot product of a vector with itself. Understanding vector magnitude is essential for normalization and for calculating adjustments during the Gram-Schmidt Process to achieve orthogonality and orthonormality.
💡Summation Notation
Summation notation is a concise way to represent the sum of a sequence of terms, indicated by the sigma symbol (∑). The video uses summation notation to simplify the expression of the Gram-Schmidt Process, particularly when subtracting the projections of the original vectors onto the newly formed orthogonal vectors. This notation helps in generalizing the process for any number of vectors.
💡Dot Product
The dot product is a way of determining the inner product of two vectors in a real vector space, resulting in a scalar. It is used in the video to calculate both the projections needed in the Gram-Schmidt Process and the magnitudes of vectors for normalization. The dot product is fundamental in many operations involving vectors, such as determining angles between vectors, vector lengths, and in this case, ensuring orthogonality and orthonormality in the process.
Highlights

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Transcripts

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