You need to use the definition of the direct sum $X\oplus Y$. The question tells you which subspaces to use for $X$ and $Y$, so what do you have to check in order to show that the definition of $V = X\oplus Y$ is satisfied?crypt50 said:Assume also that S + T = Iv and that S ∘ T = Ov = T ∘ S. Prove that V = X ⊕ Y where
X = range(S) and Y = range(T). I don't understand how to go about it, please help.
A linear map, also known as a linear transformation, is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. In other words, the output of a linear map is always a linear combination of the input vectors.
The domain of a linear map is the vector space from which the input vectors are taken, and the codomain is the vector space to which the output vectors belong. In the case of S and T, the domain and codomain are both the vector space V.
A linear map can be represented by a matrix or a set of equations. For example, if S is a linear map from V to itself, it can be represented by a square matrix with the same dimensions as the vector space V. The entries of the matrix correspond to how the linear map transforms each basis vector of V.
The terms linear map and linear transformation are often used interchangeably. However, some mathematicians make a distinction by using the term linear transformation to refer to a more general concept that includes maps between different vector spaces, while a linear map is specifically a map from a vector space to itself.
Two linear maps are considered equal if they produce the same output for every input vector. In other words, if S and T are both linear maps from V to itself, they are equal if and only if S(v) = T(v) for all v in V. This means that the matrices representing the linear maps must be identical.