Considere duas transformações,
\mathbf{T} e
\mathbf{S} :
\mathbf{T}: \mathbb{R}^{5} \longrightarrow \mathbb{R}^{5} \text { com }
\text { I) } \mathbf{T}(1,0,0,0,0)=(1,0,0,0,0), \text { II) } \mathbf{T}(0,2,0,0,0)=(2,2,0,0,0)
III)
\mathbf{T}(0,0,3,0,0)=(3,3,3,0,0), \mathbf{I V}) \mathbf{T}(0,0,0,4,0)=(4,4,4,4,0), \mathbf{V}) \mathbf{T}(0,0,0,0,5)=(5,5,5,5,5)
\begin{array}{lclc} \mathbf{S}: & \mathbb{R}^{5} & \longrightarrow & \mathbb{R}^{4} \\\ & (\mathbf{x}, \mathbf{y}, \mathbf{z}, \mathbf{w}, \mathbf{t}) & \longmapsto\left(\mathbf{x}, \sqrt[3]{\mathbf{x}^{3}+\mathbf{y}^{3}+\mathbf{z}^{3}+\mathbf{w}^{3}}, \sqrt[3]{\mathbf{x} \cdot \mathbf{y} \cdot \mathbf{z}}, \sqrt[5]{\mathbf{x} \cdot \mathbf{y} \cdot \mathbf{z} \cdot \mathbf{w} \cdot \mathbf{t}}\right) \end{array}
a) (0.75) Sabendo
\mathbf{T} é linear, determine
\mathbf{T}\left(\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3}, \mathbf{x}_{4}, \mathbf{x}_{5}\right) .
b) (0.75) Verificar que
\mathbf{S} não é uma transformação linear usando a caracterização :
\mathbf{S} \overrightarrow{\mathbf{v}}=\mathbf{A} \cdot \overrightarrow{\mathbf{v}}
c) (0.75) Verificar que
\mathbf{S} não é uma transformação linear pela definição .
d) (0.75) Calcular o núcleo de
\mathbf{T}: \mathbf{N u c}(\mathbf{T}) .
e) (0.75) Calcular a imagem de
\mathbf{T}: \mathbf{I m}(\mathbf{T}) .
f) (0.75) Verificar se
\mathbf{T} é injetiva .
g) (0.5) Verificar se
\mathbf{T} é sobrejetiva .
h) (0.5) Determine
\mathbf{T}^{-1} , se existir .
i) (1.0) Determinar uma base para
\operatorname{Nuc}(\mathbf{T}) e
\operatorname{Im}(\mathbf{T}) .
j) (1.0) Determinar
[\mathbf{T}]_{\beta}^{\alpha}, \operatorname{com} \alpha=\{(1,0,0,0,0),(0,2,0,0,0),(0,0,3,0,0),(0,0,0,4,0),(0,0,0,0,5)\} e
\beta=\{(1,0,0,0,0),(0,1,0,0,0),(0,0,1,0,0),(0,0,0,1,0),(0,0,0,0,1)\}