Discover published sets by community

Vector addition is the operation of adding two vectors together into a third vector, which is the graphical equivalent of placing the tail of the second vector at the head of the first. Example: $\vec{u} + \vec{v}$ where $\vec{u} = (1,2)$ and $\vec{v} = (3,2)$.

The determinant is a scalar value that is a function of the entries of a square matrix. It can be viewed as the scaling factor of the transformation described by the matrix, and its sign indicates the orientation. Example: The determinant of $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is $ad - bc$.

The rank of a matrix is the dimension of the vector space spanned by its columns (or rows), which is the same as the maximum number of linearly independent column (or row) vectors in the matrix. Example: Matrix $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}$ has rank 2.

The column space of a matrix is the set of all possible linear combinations of its column vectors. Example: For matrix $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, its column space is the entire $\mathbb{R}^2$ plane.

A subspace is a subset of a vector space that is a vector space in its own right under the same addition and scalar multiplication. Example: The set of all vectors in $\mathbb{R}^3$ that lie on a given plane through the origin is a subspace.

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Example: A 2x2 matrix:

A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The main diagonal itself may have non-zero elements. Example:

The span of a set of vectors is the set of all linear combinations of those vectors. Example: Vectors $\vec{u}$ and $\vec{v}$ span a plane in $\mathbb{R}^3$ if any vector in that plane can be written as $a\vec{u} + b\vec{v}$.

The dot product of two vectors is an algebraic operation that returns a scalar and is the product of their magnitudes and the cosine of the angle between them. Example: For $\vec{u} = (1,2)$ and $\vec{v} = (3,2)$, the dot product is $1\cdot3 + 2\cdot2 = 7$.

An eigenvector is a non-zero vector that changes only by a scalar factor when a linear transformation is applied to it, corresponding to its eigenvalue. Example: For $A=\begin{pmatrix} 4 & 0 \\ 0 & 5 \end{pmatrix}$, the vector $(1,0)$ is an eigenvector.

The cross product of two vectors in three-dimensional space is a vector that is perpendicular to both and has a magnitude equal to the area of the parallelogram that the vectors span. Example: For $\vec{u} = (1,0,0)$ and $\vec{v} = (0,1,0)$, the cross product is $\vec{u} \times \vec{v} = (0,0,1)$.

The Gram-Schmidt process is a method for orthogonalizing a set of vectors in an inner product space, leading to an orthogonal set. Example: Applying it to vectors $(1,1,0)$ and $(1,0,1)$ in $\mathbb{R}^3$ will result in orthogonal vectors.

A set of vectors is linearly independent if no vector in the set is a linear combination of the others. Example: In $\mathbb{R}^2$, the vectors $\vec{i} = (1,0)$ and $\vec{j} = (0,1)$ are linearly independent.

An eigenvalue of a matrix is a scalar $\lambda$ such that there exists a non-zero vector $\vec{v}$ for which $A\vec{v} = \lambda\vec{v}$, where $A$ is the matrix. Example: For matrix $A=\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$, 2 and 3 are its eigenvalues.

A linear transformation is a function between two vector spaces that preserves vector addition and scalar multiplication. Example: $T(\vec{v}) = A\vec{v}$ where $A$ is a matrix and $\vec{v}$ is a vector.

Scalar multiplication refers to the multiplication of a vector by a single number, or scalar, scaling its magnitude without changing its direction (in the absence of a rotation). Example: $3\times\vec{v}$ scales the vector $\vec{v}$ by the factor of 3.

A vector is an element of a vector space, which may be thought of as a quantity with both magnitude and direction. Example: The vector $\vec{v} = (3, 4)$ represents a direction and distance in 2-dimensional space.

A basis of a vector space is a set of linearly independent vectors that span the entire space. Example: The set $\{(1,0), (0,1)\}$ forms a basis for $\mathbb{R}^2$.

The null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in the zero vector. Example: For matrix $A=\begin{pmatrix} 1 & 2 \\ 3 & 6 \end{pmatrix}$, any scalar multiple of the vector $(-2,1)$ is in the null space.