# What exactly is a unique subspace?

I am trying to solve a problem that is asking for what conditions that make $$$$ unique. It does not explain what <> represents, but I think it means the column space. A is a matrix where its columns are a subset of the k largest eigenvectors of B.

I do not understand what it means for the column space to be unique. But if I had to guess, I would take it to mean that A is full rank, i.e., the subset of eigenvectors that form A are linearly independent.

Does anyone have a different interpretation?

Edit: Actually I just confirmed that $$$$ is the column space of A.

The column space is unique as it is defined as the smallest (wrt inclusion) subspace that contains all the columns of $$A$$ as elements. In this case it also is the set of all linear combinations of columns of $$A$$, which is minimal (as we can check it is a subspace and any subspace containing the columns must contain their linear combinations).
• Thanks. So if $A$ is full rank, then the column space of A is unique? – Iamanon Nov 20 '18 at 21:44
• @Iamanon It's always unique, but with full rank it is equal to the maximal subspace (the whole vector space the columns lie in), so $\mathbb{R}^n$ in the case of a real-valued $n \times m$ matrix. – Henno Brandsma Nov 20 '18 at 21:47
• Actually, $<A>$ is defined as the column SPAN (not space) of A. The column space is unique., but I think the column span need not be composed of the span of linearly independent columns. For example, if A is rank 3. Then we can still say that the column span is the span of the 3 columns of A. But this would not be the column space of A. The column space of A would be the span of the two linearly independent columns. – Iamanon Nov 20 '18 at 21:55