In this paper, we proposed an auto-encoder model of observation-wise linear transformation to reduce the dimensionality of data. While nonlinear models can reduce the dimensionality more effectively than linear models, such as the principal component analysis, the non-linear methods can hardly provide a simple linear relationship between the original and the dimensionally reduced data. The proposed model overcomes this difficulty while maintaining the effectiveness of the dimensionality reduction. We assessed the proposed model and compared with PCA and a typical auto-encoder model in terms of the loss function and the degree of reconstruction of the original data. By applying the proposed method to a public data of MNIST and Fashion-MNIST, we showed the effectiveness in the dimensionality reduction and relationship between the original data to the reduced data.