A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication, mapping lines to lines or points through the origin. These transformations can be represented by matrices, making them fundamental in solving systems of linear equations and understanding geometric transformations in higher dimensions.

Linear Transformation

linear transformation

function between vector spaces

Vector addition is a fundamental operation in vector algebra that combines two or more vectors to produce a resultant vector. It follows the triangle or parallelogram law, ensuring that the resultant vector maintains both magnitude and direction based on the components of the original vectors.

vector addition

Scalar multiplication involves multiplying a vector by a scalar, resulting in a new vector that is scaled in magnitude but retains the same direction unless the scalar is negative, which reverses the direction. This operation is fundamental in linear algebra and is used to scale vectors in various applications, such as physics and computer graphics.

scalar multiplication

mapping lines to lines

points through the origin

Transformations refer to operations that alter the form, appearance, or structure of something, often used in mathematics and science to describe changes in position, shape, or size while preserving certain properties. They are fundamental in understanding symmetry, geometry, and the behavior of functions and systems across various domains.

transformations

Matrices are rectangular arrays of numbers or symbols arranged in rows and columns, used to represent and manipulate linear transformations and systems of linear equations. They play a crucial role in various fields such as mathematics, physics, computer science, and engineering, providing a compact way to handle complex calculations and data structures.

matrices

Systems of linear equations consist of two or more linear equations involving the same set of variables, and their solutions represent the points where these equations intersect in n-dimensional space. Solving these systems can be achieved through methods such as graphing, substitution, elimination, or using matrix operations like Gaussian elimination or Cramer's Rule.

systems of linear equations

Geometric transformations involve changing the position, size, and orientation of shapes in a coordinate plane while preserving certain properties. They are fundamental in fields like computer graphics, robotics, and physics, where manipulating spatial objects is crucial.

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