Linear Algebra: Orthogonality and Projections

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Questions and Answers

What is the key purpose of the Gram-Schmidt process?

To decompose a matrix into its singular values and corresponding vectors.
To find the projection of a vector onto another vector.
To transform a set of linearly independent vectors into an orthonormal set. (correct)
To determine the eigenvalues and eigenvectors of a matrix.

In the context of Latent Semantic Analysis (LSA), what is the primary function of Singular Value Decomposition (SVD)?

To normalize the vectors in the document-term matrix.
To calculate the dot product between vectors.
To determine the eigenvectors of the document-term matrix.
To reduce the dimensionality of the document-term matrix. (correct)

Which of the following statements accurately describes the relationship between the projection of u onto v and the vector (u - projvu)?

Both vectors are parallel to each other.
Both vectors are orthogonal to each other. (correct)
Both vectors point in the same direction.
Both vectors have the same magnitude.

Which of these scenarios is NOT directly related to the concept of orthogonality?

Checking if a set of vectors form a basis in linear algebra. (A)

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In Latent Semantic Analysis (LSA), how is the semantic relationship between terms and documents captured?

By considering the co-occurrence of terms across multiple documents. (B)

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What is the primary benefit of using a reduced dimensionality space in Latent Semantic Analysis (LSA)?

All of the above. (D)

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Why is the dot product of two orthogonal vectors always zero?

Because they are perpendicular, the cosine of their angle is zero. (C)

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Which of the following statements is TRUE regarding the Gram-Schmidt process?

It preserves the linear span of the original set of vectors. (B)

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Flashcards

Orthogonality

Condition where two vectors are perpendicular, having a dot product of zero.

Dot Product

Mathematical operation that measures the angle between two vectors, determined by multiplying their magnitudes and cosine of the angle.

Projection of a Vector

The component of one vector in the direction of another, calculated using a specific formula.

Gram-Schmidt Process

Algorithm to convert a set of linearly independent vectors into an orthonormal set.

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Eigenvalues and Eigenvectors

Pairs where the eigenvalue scales the eigenvector without changing its direction in a linear transformation.

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Document-Term Matrix

Matrix where each row indicates a document, each column indicates a term, and cells show term frequency.

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Singular Value Decomposition (SVD)

Mathematical technique used to reduce the dimensionality of a matrix, capturing essential relationships.

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Latent Semantic Analysis (LSA)

Process that represents documents and terms in high-dimensional space, highlighting semantic relationships.

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Study Notes

Orthogonality

Orthogonal vectors are vectors that are perpendicular to each other. Their dot product is zero.
Two vectors are orthogonal if their angle is 90 degrees.
A set of vectors are orthogonal if each pair of vectors in the set is orthogonal.

Projections

The projection of a vector u onto a vector v is a vector that lies along v.
The projection of u onto v is given by the formula: proj_vu = ((u ⋅ v) / ||v||²) * v
The projection represents the component of u that lies in the direction of v.
The vector (u-proj_vu) is orthogonal to v.

Gram-Schmidt Process

The Gram-Schmidt process is an algorithm for orthonormalizing a set of vectors.
It transforms a set of linearly independent vectors into an orthonormal set.
Steps:
- Normalize the first vector in the set.
- Subtract the projection of the second vector onto the first normalized vector.
- Normalize the resulting vector.
- Repeat for subsequent vectors, subtracting projections onto all previously orthonormalized vectors.

Linear Algebra Concepts

Vectors: Quantities with both magnitude and direction.
Matrices: Arrays of numbers arranged in rows and columns.
Linear Transformations: Transformations that preserve lines and the origin.
Systems of Linear Equations: A set of equations involving linear combinations of variables.
Eigenvalues and Eigenvectors: A scalar (eigenvalue) and vector (eigenvector) pair that satisfy a specific equation.

Latent Semantic Analysis (LSA)

LSA is a technique used for information retrieval that utilizes a mathematical model.
LSA works by representing documents and terms as vectors in a high-dimensional space.
This representation emphasizes semantic relationships between words and documents based on the co-occurrence of terms.
Key Concepts:
- Document-Term Matrix: Each row represents a document, each column represents a term, and the value in each cell represents the frequency of a term in that document.
- Singular Value Decomposition (SVD): Used to reduce dimensionality of the document-term matrix. This captures important relationships between terms and documents.
- Reduced Dimensionality Space: A lower-dimensional space can be seen as a representation capturing semantic information. Documents and terms are projected onto this space.
Applications of LSA:
- Information retrieval
- Text summarization
- Topic modeling
- Similarity search