Question

# Determine whether the statement is true or false, and justify your answer. If there is a nonzero vector in the kernel of the matrix operator$T_A: R^n→R^n$then this operator is not one-to-one.

Solution

Verified

The statement is true.

By Theorem 4.10.1 the kernel of a matrix operator $T_A:\mathbb R^n\rightarrow\mathbb R^n$ is one-to-one if and only if the kernel of $T_A$ is $\left\{\textbf{0}\right\}$. Therefore if there is a nonzero vector in the kernel of a matrix operator $T_A:\mathbb R^n\rightarrow\mathbb R^n$, then this operator is not one-to-one.

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