Linear Algebra

Study of vectors, matrices, and linear transformations.

Vectors and Their OperationsMatrices and Matrix ArithmeticLinear Transformations
Determinants and InversesEigenvalues and EigenvectorsVector Spaces and Subspaces
Computer Graphics and AnimationData Science and Machine LearningRobotics and Engineering
Breaking Down Word ProblemsChecking Your Solutions with InversesVisualizing Problems to Find Solutions
Vectors and Their Op...Matrices and Matrix ...Linear Transformatio...Determinants and Inv...Eigenvalues and Eige...Vector Spaces and Su...Computer Graphics an...Data Science and Mac...Robotics and Enginee...Breaking Down Word P...Checking Your Soluti...Visualizing Problems...
Basic Concepts

Matrices and Matrix Arithmetic

What is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. Think of it as a table that helps organize numbers or equations!

Example of a matrix: \[ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \]

Matrix Operations

  • Addition: You can add matrices of the same size by adding their corresponding elements.
  • Multiplication: Multiplying matrices is a bit trickier. You multiply rows by columns and add the results.
  • Transpose: Switch the rows and columns of a matrix.

Why Learn Matrices?

Matrices are used to solve systems of equations, transform images, and more!

Try It Out

  • Add: \[ \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} \]
  • Multiply: \[ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix} \]

Examples

  • Adding two \( 2 \times 2 \) matrices element-wise.

  • Multiplying a matrix by the identity matrix leaves it unchanged.

In a Nutshell

Matrices are grids of numbers that you can add, multiply, and use to solve equations.

Key Terms

Matrix
A rectangular array of numbers organized in rows and columns.
Transpose
An operation that flips a matrix over its diagonal, switching rows and columns.
Identity Matrix
A special matrix that acts like 1 in multiplication.
Next
Linear Transformations