Computational Linear Algebra Vocabulary

Vocabulary flashcards generated from Computational Linear Algebra lecture notes.

Embedding

A transformation from strings to high dimensional vectors, typically >100D, learned by algorithms by observing large amounts of text.

Vector Databases

Searching for nearest neighbours in a vector space of embeddings.

Vectors

Ordered tuples of real numbers with a fixed dimension.

Length-3 Vector

A vector that represents a spatial position in Cartesian coordinates.

Orthogonal

Independent, or, geometrically speaking at 90°.

Vector space

A vector of given dimension lies in a vector space

Norm

Allows the length of vectors to be measured.

Inner Product

Allows the angles of two vectors to be compared; the inner product of two orthogonal vectors is 0.

Topological Vector Space

Space is continuous, and it makes sense to talk about vectors being “close together” or a vector having a neighbourhood around it.

Inner Product Space

Space where you can talk about the angle between two vectors.

Vectors

A lingua franca for data that can be composed, compared and weighted.

Vector Data

Each row that represents an “observation”.

Geometric Data

Modern computer game or 3D rendering engine.

GPUs

Evolved from devices designed to do these geometric transformations extremely quickly.

Feature Vectors

Encodes data in vector space, to output feature vectors.

k nearest neighbours

Involves some training set of data, which consists of pairs : a feature vector and a label

Mean Vector

The sum of the vectors multiplied by 1/N.

Mean vector

The geometric centroid of a set of vectors and can be thought of as capturing “centre of mass” of those vectors.

Variance

The sum of squared differences of each element from the mean of the vector.

Standard Deviation

The square root of the variance and is more often used because it is in the same units as the elements.

Covariance

The average squared difference of each column of data from the average of every column.

Covariance Ellipses

A summary of high-dimensional data that represents a (inverse) transform of a unit sphere to an ellipse covering the data.

Diagonal entries of a matrix

Are important “landmarks” in the structure of a matrix; elements are often referred to as being “diagonal” or “off-diagonal” terms.

Diagonal Matrices

Matrices which are all zero except for a single diagonal entry that represent an independent scaling of each dimension.

Identity Matrix

Is denoted and is a square matrix, where all values are zero except 1 along the diagonal.

Zero Matrix

All zeros, and is defined for any matrix size; maps all vectors onto the zero vector (the origin).

Square Matrix

If it has size ; apply transformations within a vector space; mappings from dimensional space to dimensional space

Triangular Matrix

Has non-zero elements only above (upper triangular) or below the diagonal (lower triangular), inclusive of the diagonal. That represent particularly simple to solve sets of simultaneous equations.