Linear Algebra Principles and Matrix Properties

This flashcard set covers core concepts from linear algebra including matrix inversion, Cramer's rule, vector space properties, eigenvalues, and inner products as detailed in the lecture notes.

Adjoint Matrix ($adj(A)$)

The transpose of the cofactor matrix $C$, calculated as $adj(A) = C^T$.

Inverse of a Matrix ($A^{-1}$)

For a matrix $A$, it is defined as $A^{-1} = \frac{1}{det(A)} adj(A)$.

Cramer’s Rule

A technique for solving systems of linear equations using determinants, where variables are found via ratios such as $x = \frac{D_x}{D}$, $y = \frac{D_y}{D}$, and $z = \frac{D_z}{D}$.

Linearly Independent Vectors

A set of vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n$ where the equation $k_1\mathbf{v}_1 + k_2\mathbf{v}_2 + \dots + k_n\mathbf{v}_n = \mathbf{0}$ is satisfied only by the trivial solution $k_1 = k_2 = \dots = k_n = 0$.

Linearly Dependent Vectors

A set of vectors where at least one vector can be expressed as a linear combination or scalar multiple of the others, or if the set contains the zero vector.

Basis

A set of linearly independent vectors that span a specific solution space or subspace.

Dimension ($dim$)

The number of vectors contained within a basis for a given space.

Rank ($Rank(A)$)

The number of non-zero rows present in the reduced row echelon form ($R$) of a matrix $A$.

Nullity ($Nullity(A)$)

The value determined by subtracting the rank from the total number of columns in the matrix ($Nullity = \text{number of columns} - Rank$).

Characteristic Equation

The equation defined by $det(\lambda I - A) = 0$, used to determine the eigenvalues of a matrix $A$.

Eigenvalues ($\lambda$)

The scalar values produced by solving the characteristic equation $det(\lambda I - A) = 0$.

Eigenvectors

Non-zero vectors $\mathbf{x}$ that satisfy the system $(\lambda I - A)\mathbf{x} = \mathbf{0}$ for a given eigenvalue $\lambda$.

Diagonalization ($A = PDP^{-1}$)

A process where a matrix $A$ is decomposed into a matrix of eigenvectors $P$ and a diagonal matrix of eigenvalues $D$.

Orthogonal Matrices

Two matrices $\mathbf{u}$ and $\mathbf{v}$ are considered orthogonal with respect to an inner product if $\langle \mathbf{u}, \mathbf{v} \rangle = 0$.

Standard Inner Product on $M_{22}$

An inner product for $2 \times 2$ matrices defined as $\langle \mathbf{u}, \mathbf{v} \rangle = tr(U^T V)$.

Norm ($\left\| U \right\|$)

The magnitude of a vector or matrix, often calculated as the square root of the sum of the squares of its entries.

Distance ($d(U, V)$)

The magnitude of the difference between two vectors, defined as $\left\| U - V \right\|$.

Weighted Euclidean Inner Product

An inner product in $\mathbb{R}^n$ defined with specific weights for each component, such as $\langle \mathbf{u}, \mathbf{v} \rangle = w_1 u_1 v_1 + w_2 u_2 v_2$.