MathAlgebraLinear algebraLinear operators

Introduction to linear operators

A linear operator or not

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Consider a vector space $\text{Mat}_2(\mathbb{R}) \ -$ a vector space of $2 \times 2$ -matrices with real coefficients. Suppose we have a mapping $\mathcal{A} \colon \text{Mat}_2(\mathbb{R}) \to \text{Mat}_2(\mathbb{R})$ , which is defined by the following rule: for any matrix $X \in\text{Mat}_2(\mathbb{R})$

$\mathcal{A}(X) = \begin{pmatrix} 1 & -1 \\ 0 & 2 \end{pmatrix} \cdot X \cdot \begin{pmatrix} 2 & -1 \\ 0 & 3 \end{pmatrix} + X.$ Is $\mathcal{A}$ a linear operator?

Select one option from the list

This is an operator, but it is not linear.

Nothing can be said, it depends on a matrix X.

Yes, this is a linear operator.

This is not a mapping.

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