Suppose A is an n ×n matrix with the property that the equation A 𝐱=0 has only the trivial solut

step 1 we want to show that if a x equals 0 has only the trivial solution, trivial solution, then we wa

Suppose A is an n ×n matrix with the property that the...

Suppose $A$ is an $n \times n$ matrix with the property that the equation $A \mathbf{x}=\mathbf{0}$ has only the trivial solution. Without using the Invertible Matrix Theorem, explain directly why the equation $A \mathbf{x}=\mathbf{b}$ must have a solution for each $\mathbf{b}$ in $\mathbb{R}^{n}$ .

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Key Concepts

Trivial Kernel

When the equation A x = 0 has only the trivial solution, it means that the null space of A contains only the zero vector. This indicates that the associated linear transformation is injective, as no nonzero vector is sent to the zero vector.

Linear Independence and Basis

Because A is an n × n matrix with a trivial null space, its columns must be linearly independent. In an n-dimensional space, n linearly independent vectors form a basis. Having a basis means that every vector in ?? can be uniquely expressed as a linear combination of the columns of A.

Properties of Finite-Dimensional Spaces

In finite-dimensional vector spaces, if a linear transformation is injective and the domain and codomain have the same dimension, then the transformation is also surjective. This guarantees that for every vector b in ?? there exists a vector x such that A x = b.

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