Prove that the given rule is a linear transformation, or show why it isn’t.
(a) T : $$\mathbb{R}^\infty \rightarrow \mathbb{R}^\infty$$ T is the rule “shift to the right”: T(x1, x2, x3, . . .) = (0, x1, x2, x3, . . .).

(b) T : M2×3 $$(\mathbb{R}) \rightarrow \mathbb{R}^2$$ T sends the matrix M to Mw where ~w = (1, 2, 3) (the output is a vector in $$\mathbb{R}^2$$.

(c) T : $$W _{2} \rightarrow \mathbb{R}^2$$, T(x, y) = (x, y).

(d) T : $$W _{2} \rightarrow \mathbb{R}^2$$, T(x, y) = (ln(x), ln(y)).

Question

Prove that the given rule is a linear transformation, or show why it isn’t.
(a) T : $$\mathbb{R}^\infty \rightarrow \mathbb{R}^\infty$$  T is the rule “shift to the right”: T(x1, x2, x3, . . .) = (0, x1, x2, x3, . . .).

(b) T : M2×3 $$(\mathbb{R}) \rightarrow \mathbb{R}^2$$ T sends the matrix M to Mw where ~w = (1, 2, 3) (the output is a vector in $$\mathbb{R}^2$$.

(c) T : $$W _{2} \rightarrow \mathbb{R}^2$$, T(x, y) = (x, y).

(d) T : $$W _{2} \rightarrow \mathbb{R}^2$$, T(x, y) = (ln(x), ln(y)).

anonymous · Answer

(e) T : $$C^\infty (\mathbb{R}) \rightarrow \mathbb{R},  T(f) = \int\limits_{1}^{3} f \times \sin (x)dx$$