1 is obvious:
Since it is a linear transformation, B0 = 0, and the result of Bx is of course still in X.
2 is also obvious:
For example, all m×n matrices on a field of real numbers form a linear space, and all symmetric or singular matrices in it are its subspaces.
In three-dimensional space, any plane that crosses the origin can be considered a linear subspace. The points (vectors) on the plane form a linear subspace under addition and multiplication.
Linear operators are an important concept in linear space that represents linear transformations A linear operator is a mapping from one linear space to another, maintaining addition and number multiplication operations. For example, matrix multiplication represents a linear transformation that maps one vector space to another.
Any plane that crosses the origin in 3D space can be considered a linear subspace of 3D space. Then the space formed by the vectors in any plane is still this plane, and it still belongs to this plane after linear transformation.
The domain of linear operator B is D(B) and the value range is R(B), both of which are X.
Therefore, if x ∈ D(B), then Bx gets a part of R(B) and BBx gets part of D(B), because BBx can be applied to Bx again by the action of B. Since both D(B) and R(B) are X, BBx does fall within the scope of Bx.
4 is also obvious.
Suppose L is the space of the eigenvector corresponding to the eigenvalues of B, so L is the invariant subspace of B
By definition, a subspace W is an invariant subspace of a linear transformation T, if and only if there is T(x)∈W for an arbitrary vector x∈W.
对于特征向量张成的空间L，设L是由特征向量x1, x2, ..., xn张成的空间，满足B(xi) = λixi，其中λi是xi对应的特征值。 对于任意向量α∈L，可以表示为α=c1x1+c2x2+...+cnxn，其中ci是常数。 由于每个xi都是B的特征向量，有B(xi)=λixi，因此B(α)=B(c1x1+c2x2+...+cnxn)=c1λ1x1+c2λ2x2+...+cnλnxn，这仍然在L中。 因此，L是B的不变子空间。