Algebra of Linear Transformations A linear transformation is a function that takes vectors from one vector space to another vector space and acts in a s...
Algebra of Linear Transformations
A linear transformation is a function that takes vectors from one vector space to another vector space and acts in a specific way on each vector, known as its linearity.
In linear algebra, we often consider linear transformations as matrices. A matrix is a rectangular array of numbers that represents a linear transformation.
T(u + v) = T(u) + T(v)
T(cu) = cT(u)
Determinant property:
The determinant of a linear transformation is a scalar that can be calculated from the matrix.
If the determinant of a matrix is zero, the linear transformation is not invertible, which means it cannot be used to transform vectors between vector spaces.
Scaling: T(u) = cu for a constant c.
Rotation: T(u) = rotation about the origin through an angle θ.
Projection onto a subspace: T(u) = u · v for a vector v.
By understanding linear transformations, we can perform various operations on vectors, such as addition, multiplication, and composition of transformations. These operations can be expressed using matrices, making linear algebra a powerful tool for solving problems involving linear transformations