Suppose 𝐛 ∈img A and ker A={0}. Prove that 𝐱^*=A^+ 𝐛 is the unique solution to the linear system

step 1 For this exercise, we have vector x sub h equal to P .R. OJ sub kernel of a x vector is an eleme

Suppose 𝐛 ∈img A and ker A={0}. Prove that 𝐱^*=A^+ 𝐛 is the...

Suppose $\mathbf{b} \in \operatorname{img} A$ and $\operatorname{ker} A=\{0\}$. Prove that $\mathbf{x}^*=A^{+} \mathbf{b}$ is the unique solution to the linear system $A \mathrm{x}=\mathrm{b}$. What if $\operatorname{ker} A \neq\{0\}$ ?

Key Concepts

Injectivity and Uniqueness of Solutions

Injectivity of a matrix, characterized by a trivial kernel (i.e., ker A = {0}), guarantees that each output is produced by a unique input. In the context of solving linear systems, this means that if b is in the image of A and A has a trivial kernel, the solution given by the pseudoinverse is the unique solution. This concept is crucial in understanding why certain systems have unique solutions while others may have infinitely many solutions.

Minimal-Norm Solutions and Overdetermined Systems

When the kernel of a matrix is nontrivial, multiple solutions exist for a given linear system. In these cases, the Moore-Penrose pseudoinverse provides the unique solution that has the smallest norm (i.e., the minimal-norm solution). This property is particularly important in contexts where among all possible solutions, one seeks the one that does not introduce extraneous effects, ensuring stability and optimality in applications such as least-squares problems.

Moore-Penrose Pseudoinverse

The Moore-Penrose pseudoinverse is a generalization of the matrix inverse that can be applied even when a matrix is not square or not invertible in the usual sense. It provides a way to define a unique, well-behaved solution that is optimal in the least-squares sense, making it an essential tool for solving linear systems and for data fitting. This pseudoinverse satisfies several important properties, such as A A? A = A and A? A A? = A?, which are used to verify its correctness and to ensure that it yields the minimal-norm solution among all possible solutions when one exists.

Kernel (Nullspace) of a Matrix

The kernel or nullspace of a linear transformation describes the set of all input vectors that map to the zero vector. Its properties are fundamental in determining the uniqueness of solutions to a linear system; specifically, if the kernel contains only the zero vector, the transformation is injective, which implies that any solution to the equation is unique. Conversely, a nontrivial kernel leads to the possibility of multiple solutions, as any vector in the kernel can be added to a particular solution to generate another solution.

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