Suppose we have six matrices (whose sizes are chosen in such a way that all the matrix multiplications are defined in inequality (1) below) given as follows $$\mathbb{H_{31},V_1,H_{32},V_2,H_{41},H_{42}}.$$ Further assume that each of $\mathbb{H_{31},H_{32},H_{41},H_{42}}$ is full rank and all of their individual columns are independent of each other. This means that if we concatenate $\mathbb{H_{31},H_{32},H_{41},H_{42}}$ then the resulting matrix will be full rank. In this situation is the following inequality always true? $$\dim(\mathbb{V_1})\geq \dim(\mathbb{H_{31}V_1\cap H_{32}V_2})+\dim(\mathbb{H_{41}V_1\cap H_{42}V_2}).\tag{1}$$
2026-03-25 20:42:31.1774471351
Is the following inequality always true for matrices?
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