A class of splitting alternating algorithms is proposed for finding the sparse solution of linear systems with concatenated orthogonal matrices. Depending on the number of matrices concatenated, the proposed algorithms are classified into the two-block splitting alternating algorithm (TSAA) and the multi-block splitting alternating algorithm (MSAA). These algorithms aim to decompose a large-scale linear system into two or more coupled subsystems, each significantly smaller than the original system, and then combine the solutions of these subsystems to produce the sparse solution of the original system. The proposed algorithms only involve matrix-vector products and reduced orthogonal projections. It turns out that the proposed algorithms are globally convergent to the sparse solution of a linear system if the matrix (along with the sparsity level of the solution) satisfies a coherence-type condition. Numerical experiments indicate that the proposed algorithms are very promising and can quickly and accurately locate the sparse solution of a linear system with significantly fewer iterations than several mainstream iterative methods.