Linear Transformations
A linear transformation is a function that maps a vector space onto itself, and preserves the operations of vector addition and scalar multiplication
A linear transformation is a function that maps a vector space onto itself, and preserves the operations of vector addition and scalar multiplication. In simple terms, it is a transformation that preserves lines, planes, and their relative distances.
Formally, let V and W be vector spaces. A function T: V -> W is a linear transformation if it satisfies two properties:
1. Additivity: T(u + v) = T(u) + T(v) for all u, v in V.
This means that the transformation of the sum of two vectors is equal to the sum of the transformations of the individual vectors.
2. Homogeneity of degree one: T(cu) = cT(u) for all c in the field of scalars and all u in V.
This means that the transformation of a scalar multiple of a vector is equal to the same scalar multiple of the transformation of the original vector.
Linear transformations can be represented by matrices. Let’s say we have a linear transformation T: R^n -> R^m. If we choose standard bases for the vector spaces, then T can be represented by an m x n matrix A, where each column of A represents the transformation of a basis vector from R^n to R^m.
For example, let’s consider the linear transformation T: R^2 -> R^2 defined as T(x, y) = (2x + y, 3y). To find the matrix representation for this transformation, we can apply T to the standard basis vectors of R^2.
T(1, 0) = (2(1) + 0, 3(0)) = (2, 0)
T(0, 1) = (2(0) + 1, 3(1)) = (1, 3)
The matrix representation A for the linear transformation T is:
A = | 2 1 |
| 0 3 |
To apply this transformation to any vector in R^2, we simply multiply the vector by the matrix A. For example, let’s apply the transformation to the vector (3, 4):
A * (3, 4) = | 2 1 | * | 3 | = | 10 |
| 4 | | 12 |
So, the image of (3, 4) under the linear transformation T is (10, 12).
Linear transformations have many important applications in various fields, including computer graphics, data analysis, and physics. They provide a way to represent and analyze transformations of vectors in a mathematically rigorous manner.
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