In this thesis we examine the geometry of the singular values decomposition (SVD) of a matrix or a linear operator. The singular values of the matrix \(A\) are the square roots of the eigenvalues of the Hermitian symmetric, positive definite matrix \(A^t A\). The key fact is that \(A^t A\) is diagonalized in an orthonormal basis and the action of \(A\) on the eigenvectors of \(A^t A\) preserves orthogonality. In SVD, matrix \(A\) of type \((m,n)\) is factorized as a product of three matrices, of which the outer one are orthogonal/unitary, and the middle one is of the same type as \(A\) and contains a diagonal submatrix with singular values on the diagonal. Geometrically, this factorization means the composition, respectively, of an isometry, which is used for transition to the orthonormalized base of eigenvectors of \(A^t A\), then scaling along the directions of these vectors, and finally another isometry. In the Euclidean plane, SVD is applied to the bijective affine transformation \(A\) and its action on the unit circle. The image is an ellipse, defined by any pair of conjugate diameters as the image of a pair of orthogonal diameters of a circle. The main axes of the ellipse are obtained by the Rytz construction, and the lengths of the semi-axes of the ellipse are equal to the singular values of the operator \(A\). Another geometric application of SVD refers to oblique axonometry, that is, the parallel projection of Euclidean space onto a plane, in which the image of an orthonormal base of space is given. Using a similar procedure, projections of the apparent and real contours of the unit sphere are constructed.