Anomaly Detection (AD) has recently become an important application of hyperspectral images analysis. The goal of these algorithms is to find the objects in the image scene which are anomalous in comparison to their surrounding background. One way to improve the performance and runtime of these algorithms is to use Dimensionality Reduction (DR) techniques. This paper evaluates the effect of three popular linear dimensionality reduction methods on the performance of three benchmark anomaly detection algorithms. The Principal Component Analysis (PCA), Fast Fourier Transform (FFT) and Discrete Wavelet Transform (DWT) as DR methods, act as pre-processing step for AD algorithms. The assessed AD algorithms are Reed-Xiaoli (RX), Kernel-based versions of the RX (Kernel-RX) and Dual Window-Based Eigen Separation Transform (DWEST). The AD methods have been applied to two hyperspectral datasets acquired by both the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) and Hyperspectral Mapper (HyMap) sensors. The evaluation of experiments has been done using Receiver Operation Characteristic (ROC) curve, visual investigation and runtime of the algorithms. Experimental results show that the DR methods can significantly improve the detection performance of the RX method. The detection performance of neither the Kernel-RX method nor the DWEST method changes when using the proposed methods. Moreover, these DR methods increase the runtime of the RX and DWEST significantly and make them suitable to be implemented in real time applications.