Linear Algebra — Vectors & Matrices

Matrix

A rectangular array of numbers arranged in rows and columns.

Dot Product

The scalar result of multiplying corresponding entries of two vectors and summing those products.

Cross Product

A vector perpendicular to two given vectors in three-dimensional space.

Identity Matrix

A square matrix with ones on the diagonal and zeros elsewhere.

Transpose of a Matrix

A new matrix obtained by swapping rows with columns of the original matrix.

What is the determinant of a 2x2 matrix $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$?

$ad-bc$

Eigenvalue

A scalar $\lambda$ such that $A\vec{v}=\lambda \vec{v}$ for some non-zero vector $\vec{v}$.

Eigenvector

A non-zero vector $\vec{v}$ that satisfies $A\vec{v}=\lambda \vec{v}$ for some scalar $\lambda$.

Rank of a Matrix

The maximum number of linearly independent row vectors or column vectors in the matrix.

Singular Matrix

A square matrix that does not have an inverse, i.e., its determinant is zero.

Orthogonal Vectors

Two vectors with a dot product of zero.

Orthogonal Matrix

A square matrix whose rows and columns are orthonormal vectors.

Inverse of a Matrix

A matrix $A^{-1}$ such that $A{A}^{-1}=A^{-1}A=I$, where $I$ is the identity matrix.

What is the formula for the inverse of a 2x2 matrix $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$?

$\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$, assuming $ad-bc\ne 0$.

Basis

A set of linearly independent vectors that span a vector space.

Span

The set of all possible linear combinations of a given set of vectors.

Null Space

The set of all vectors $\vec{v}$ such that $A\vec{v}=\vec{0}$ for a matrix $A$.

Column Space

The set of all linear combinations of the columns of a matrix.

Gram-Schmidt Process

A method for orthogonalizing a set of vectors in an inner product space.

LU Decomposition

A factorization of a matrix into a lower triangular matrix $L$ and an upper triangular matrix $U$.

QR Decomposition

A factorization of a matrix into an orthogonal matrix $Q$ and an upper triangular matrix $R$.

Row Echelon Form

A form of a matrix where all nonzero rows are above any row of all zeros, and each leading entry of a row is to the right of the leading entry of the row above it.

Reduced Row Echelon Form

A row echelon form where every leading entry is 1 and is the only nonzero entry in its column.

What is the purpose of matrix multiplication in linear transformations?

To represent the composition of linear transformations as matrix multiplication.

What is a linear transformation?

A function between vector spaces that preserves vector addition and scalar multiplication.

Diagonal Matrix

A matrix with non-zero entries only on its main diagonal.

Trace of a Matrix

The sum of the elements on the main diagonal of a square matrix.

Symmetric Matrix

A matrix that is equal to its transpose.

Skew-Symmetric Matrix

A matrix that is equal to the negative of its transpose.

Vector

An element of a vector space, represented as an ordered list of numbers.

What does the Cayley-Hamilton theorem state?

Every square matrix satisfies its own characteristic equation.

What is the characteristic polynomial of a matrix $A$?

The polynomial $\det (A-\lambda I)$, where $\lambda$ is a scalar and $I$ is the identity matrix.

Orthogonal Projection

The projection of a vector onto a subspace using an orthogonal basis.

What is a unit vector?

A vector with a magnitude of 1.

What is the spectral theorem?

Any symmetric matrix can be diagonalized by an orthogonal matrix.