Find all left ideals of $M_n(\mathbb{K})$ where $\mathbb{K}$ is a field.
#Left ideals of matrices over a field
dreamy olive
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Casiel368
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Find all left ideals of $M_n(\mathbb{K})$ where $\mathbb{K}$ is a field.
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prisma orbit
Find all left ideals of $M_n(\mathbb{K})$ where $\mathbb{K}$ is a field.
For now only the trivial, the improper and the singular matrices come to mind
dreamy olive
Not exactly all the singular ones, because addition of singular might not be singular
Their kernels must have nontrivial intersection
prisma orbit
Not exactly all the singular ones, because addition of singular might not be sin...
Very true, I was fixated on the multiplication
dreamy olive
It does shed some light though
hard crane
the left ideals are in one to one correspondence with left submodules of the free left K module K^n
if you take any submodule M < K^n, then you have an ideal I(M) of matrices whose rows are elements of M
conversely as well, given any left ideal I of matrices, we can define a left submodule M(I) which consists of the first rows of the elements in the ideal
@dreamy olive
so it suffices to find all left submodules of K^n which shouldn't be difficult