In the area of array signal processing, direction of arrival (DoA) estimation is a widely studied topic. In this estimation process the noise subspace of the received signal covariance matrix is often utilized and obtained through numerical methods. We explicitly derive an algebraic expression of the noise subspace when the number of signal sources present is less than the number of elements of a uniform linear array (ULA). This expression of the noise subspace is then used to formulate a constrained minimization problem to obtain the DoAs of all the sources in a scene in the presence of spatially white noise of identical power. This noise subspace-based estimation (NISE) algorithm iteratively solves for each source's DoA, potentially yielding (depending on the number of iterations) lower complexity than existing DoA estimation algorithms, such as fast root-MUSIC (FRM), while exhibiting performance advantages for a low number of time samples and low signal-to-noise ratio (SNR). The convergence of NISE is then proven mathematically. In addition it is shown how NISE can readily incorporate prior knowledge into the DoA estimation process.