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Let , and be vector spaces, and let and be linear transformations. The composition
of and is the transformation given by
The matrix of a composition is the product of the matrices corresponding to the
transformations in the composition, in the same order.
Image of a linear transformation
Let and be vector spaces, and let be a linear transformation. The image of ,
denoted by , is the set
In other words, the image of consists of individual images of all vectors of .
Inverse of a linear transformation
Let and be vector spaces, and let be a linear transformation. A transformation
that satisfies and is called an inverse of . If has an inverse, is called invertible.
Kernel of a linear transformation
Let and be vector spaces, and let be a linear transformation. The kernel of ,
denoted by , is the set
In other words, the kernel of consists of all vectors of that map to in .
Linear transformation
A transformation is called a linear transformation if the following are true for all
vectors and in , and scalars .
Standard matrix of a linear transformation
Let be a linear transformation. Then the matrix
is known as the standard matrix of the linear transformation .
2024-09-06 02:10:41
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Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)