Question

Determine a matriz dos autovetores do operador linear T : R 2 → R 2 . definido por T ( x , y ) = ( 4 x + 5 y , 2 x + y )

Answer 1

Resposta:

Explicação passo-a-passo:

A matriz [T] de T(x, y) = (4x + 5y, 2x + y) ´e:

[T] = $\left[\begin{array}{cc}4&5\\2&1\end{array}\right]$

Portanto, procurar os autovalores e autovetores de T equivale

a resolver:

[T(v)] = λ.[v] ⇐⇒ [T].[v] = λ[v]

Ou seja

$\left[\begin{array}{cc}4&5\\2&1\end{array}\right]*\left[\begin{array}{c}x&y\end{array}\right]=\lambda*\left[\begin{array}{c}x&y\end{array}\right]\longleftrightarrow\left[\begin{array}{cc}4&5\\2&1\end{array}\right]*\left[\begin{array}{c}x&y\end{array}\right]= \left[\begin{array}{cc}1&0\\0&1\end{array}\right]*\left[\begin{array}{c}x&y\end{array}\right]\longleftrightarrow[T].[v] = \lambda.I.[v]\longleftrightarrow[T].[v]-\lambda.I.[v] = [0]\longleftrightarrow([T]-\lambda.I)[v]=[0]$