Linear Equations and Linear Transformations
- (Duke University - Cheng-Yu Chen)
- Linear Equations
A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation. The standard form of a linear equation in one variable is of the form Ax + B = 0. Here, x is a variable, A is a coefficient and B is constant.
Please refer to the following for more details:
- Wikipedia: Linear Equation
- Linear Transformations
In linear algebra, a linear transformation (or linear map) is a function that moves from one vector space to another. It also respects the underlying structure of each vector space. A linear transformation is also known as a linear operator or map.
A transformation is linear if it satisfies the following two properties:
- Additive: The output is the same if the numbers are added first and then transformed, or if they are transformed and then the transformations are added together.
- Scalar: For all vectors →x and all scalars k, T(k→x) = kT(→x).
Examples of linear transformations include the zero transformation and the identity transformation. The zero transformation is defined by T(→x) = →(0) for all →x. The identity transformation is defined by T(→x) = →(x).
Matrices can be used to perform a wide variety of transformations on data.
Please refer to the following for more information:
- Wikipedia: Linear Transformation
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