This segment explains a computationally efficient method for approximating matrix-vector products (Ax) using the Singular Value Decomposition (SVD). By selectively ignoring singular values below a certain threshold, the computation is significantly simplified without substantial loss of accuracy. This highlights a practical application of SVD for large-scale computations. The lecture derives the pseudoinverse (A⁺) using SVD (Singular Value Decomposition): A = UΣVᵀ. A⁺ optimizes least squares problems, providing minimum-norm solutions for underdetermined systems. A⁺ = VΣ⁺Uᵀ, where Σ⁺ inverts non-zero singular values. SVD allows efficient approximation by ignoring small singular values, enabling rank-k approximation. The Frobenius norm of A is the sum of squared singular values. SVD is computationally expensive but crucial for proving theorems and solving various linear algebra problems. This segment details the properties of the pseudoinverse (A+), explaining its behavior for square invertible matrices, overdetermined systems (least squares solution), and underdetermined systems (minimum norm least squares solution). This concisely summarizes the versatility and applications of the pseudoinverse across different matrix scenarios.