MATHEMATICS · STRUCTURE AND TRANSFORMATION

Linear Algebra

Linear algebra studies vectors, matrices and linear transformations. It is the language used to solve coupled linear equations, describe geometry in many dimensions and analyze how linear systems stretch, rotate or preserve directions.

Vectors and matrices

A vector in \(\mathbb{R}^n\) is an ordered list of \(n\) numbers. Vector addition and scalar multiplication are performed componentwise.

\[\mathbf{x}=\begin{bmatrix}x_1\\x_2\\\vdots\\x_n\end{bmatrix}.\]

An \(m\times n\) matrix represents a linear map from \(\mathbb{R}^n\) to \(\mathbb{R}^m\). Matrix-vector multiplication forms each output component as a linear combination of the input components.

\[A\mathbf{x}=\mathbf{b}.\]

Linear maps

A map \(T\) is linear when it preserves vector addition and scalar multiplication:

\[T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v}),\qquad T(c\mathbf{u})=cT(\mathbf{u}).\]

Once a basis is chosen, every linear map between finite-dimensional vector spaces can be represented by a matrix.

Systems of linear equations

A system of linear equations can be written compactly as \(A\mathbf{x}=\mathbf{b}\). Row operations preserve the solution set and lead to row-echelon form, which reveals whether the system has no solution, one solution or infinitely many solutions.

Key criterion: a square matrix \(A\) has a unique solution for every \(\mathbf{b}\) exactly when \(A\) is invertible.

Determinant and invertibility

For a square matrix, the determinant is a scalar that measures oriented volume scaling. For a \(2\times2\) matrix,

\[\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc.\]

A square matrix is invertible if and only if its determinant is nonzero. In that case, \(A\mathbf{x}=\mathbf{b}\) has the unique solution \(\mathbf{x}=A^{-1}\mathbf{b}\).

Eigenvalues and eigenvectors

A nonzero vector \(\mathbf{v}\) is an eigenvector of \(A\) when applying \(A\) changes only its scale, not its direction:

\[A\mathbf{v}=\lambda\mathbf{v}.\]

The scalar \(\lambda\) is the corresponding eigenvalue. Eigenvalues are found from the characteristic equation

\[\det(A-\lambda I)=0.\]

Eigenvectors reveal invariant directions and are central to stability analysis, differential equations, data analysis and repeated matrix transformations.

A practical workflow

Choose the representation.

Write the problem using vectors and a matrix whenever the relationships are linear.

Check dimensions.

For \(A\mathbf{x}\), the number of columns of \(A\) must equal the number of components of \(\mathbf{x}\).

Test structure.

For square matrices, rank, determinant and eigenvalues answer different questions about solvability and transformation behavior.

Interpret the result.

Do not stop at a matrix calculation: translate the solution, null space or eigenvector back into the original problem.

Connections

Linear algebra connects directly to systems of equations, multivariable calculus and differential equations.