This video describes the relationship between a direct sum decomposition and the coordinates of a vector.

Direct sum - In this video, we have the direct sum decomposition

\[V = V_1 \oplus V_2 \oplus \cdots \oplus V_n\]

where \(V_1\), \(V_2\), …, \(V_n\) are subspaces of \(V\). This means that each \(\mathbf{v}\in V\) can be expressed uniquely as

\[\mathbf{v} = \mathbf{v}_1 + \cdots + \mathbf{v}_n\]

where \(\mathbf{v}_1\in V_1\), \(\mathbf{v}_2\in V_2\), …, \(\mathbf{v}_n\in V_n\).

Each subspace has its own basis, \(B_i = \mathbf{v}_{r_1 + \cdots + r_{i-1} + 1},\ldots \mathbf{v}_{r_1 + \cdots + r_i}\). In other words, \(V_i\) has dimension \(r_i\) and so the dimension of \(V\) is \(r_1 + \cdots + r_n\).

In the video we see how the coordinates of a vector \(\mathbf{v}\in V\) decomposes in terms of the direct sum decomposition.