Linear algebra is a branch of mathematics that deals with vectors, matrices, and linear transformations. Here are some examples that demonstrate the concepts of linear algebra.

Vectors

Vectors are used to represent quantities that have both magnitude and direction. For example, velocity and force are both vectors.

  • Example 1: The velocity of a car moving east at 60 miles per hour can be represented by the vector ( \vec{v} = \begin{bmatrix} 60 \ 0 \end{bmatrix} ).

Matrices

Matrices are rectangular arrays of numbers. They are used to represent systems of linear equations and transformations.

  • Example 2: The matrix ( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} ) can be used to represent a linear transformation that scales and rotates a point in the plane.

Linear Transformations

Linear transformations are functions that map vectors to vectors. They are represented by matrices.

  • Example 3: The linear transformation ( T ) defined by the matrix ( A ) from Example 2 scales and rotates any point in the plane.

Further Reading

For more information on linear algebra, you can visit our Linear Algebra Tutorial.

Vector

Matrix

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