True or false: If 𝐱1^⋆ solves A 𝐱=𝐜, and 𝐱2^⋆ solves B 𝐱=𝐝, then 𝐱^⋆=𝐱1^⋆+𝐱2^⋆ solves (A+B) 𝐱=𝐜+

step 1 All right, so we're looking at chapter 2, section 3, question 35. Now, for part A, this is going

True or false: If 𝐱1^⋆ solves A 𝐱=𝐜, and 𝐱2^⋆ solves B 𝐱=𝐝,...

True or false: If $\mathbf{x}_1^{\star}$ solves $A \mathbf{x}=\mathbf{c}$, and $\mathbf{x}_2^{\star}$ solves $B \mathbf{x}=\mathbf{d}$, then $\mathbf{x}^{\star}=\mathbf{x}_1^{\star}+\mathbf{x}_2^{\star}$ solves $(A+B) \mathbf{x}=\mathbf{c}+\mathbf{d}$. 2.5.20. Under what conditions on the coefficient matrix $A$ will the systems in (2.34) all have a solution?

True or false: If $\mathbf{x}_1^{\star}$ solves $A \mathbf{x}=\mathbf{c}$, and $\mathbf{x}_2^{\star}$ solves $B \mathbf{x}=\mathbf{d}$, then $\mathbf{x}^{\star}=\mathbf{x}_1^{\star}+\mathbf{x}_2^{\star}$ solves $(A+B) \mathbf{x}=\mathbf{c}+\mathbf{d}$.
2.5.20. Under what conditions on the coefficient matrix $A$ will the systems in (2.34) all have a solution?

Key Concepts

Column Space and Solvability Conditions

The concept of the column space is central to solving linear systems because it comprises all possible linear combinations of the matrix’s columns. A system Ax=b is solvable if and only if the vector b lies in the column space of A. In contexts where a family of systems must be solvable for arbitrary right-hand sides, A is required to have its column space equal to the entire codomain (for instance, R^m in an m×n matrix), ensuring that every choice of b allows for a solution.

Superposition Principle and Linearity

The superposition principle in linear algebra states that if a system is homogeneous, any linear combination of its solutions is also a solution. However, when dealing with two different systems, such as A x=c and B x=d, summing individual solutions does not generally yield a solution to (A+B)x=c+d unless specific additional conditions, like particular compatibility or commutativity, are met.

Invertibility and Full Rank

A matrix is said to be invertible (or nonsingular) when it is square and has full rank, meaning all its rows (or columns) are linearly independent. In such cases, every system of the form Ax=b has a unique solution for any vector b. For non?square matrices, being of full row rank or full column rank plays a similar role in ensuring the existence or uniqueness of solutions to the associated systems.

Linear Systems and Consistency

A linear system of equations, expressed in the form Ax=b, has a solution if and only if the right?hand side vector b belongs to the column space (range) of the matrix A. This concept, known as the consistency of the system, is fundamental to determining whether solutions exist, regardless of whether the solution is unique or there are infinitely many.

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