A = ( k − 2 1 − k k ) , where k is...

gorgeousgen9487

gorgeousgen9487

Answered

2022-07-06

A = ( k 2 1 k k ) , where k is a constant
A transformation  T : R 2 R 2  is represented by the matrix A.
Find the value of k for which the line  y = 2 x  is mapped onto itself under T.

Answer & Explanation

Valeria Wolfe

Valeria Wolfe

Expert

2022-07-07Added 11 answers

The line y = 2 x is mapped to itself if we have that for each point ( x , y ) on the line T ( x , y ) also lies on the line. We needn't have T ( x , 2 x ) = ( x , 2 x ) for all x, it suffices to have T ( x , 2 x ) = ( ϕ ( x ) , 2 ϕ ( x ) ) for all x and some function ϕ.
Having this in mind, we look at T ( x , 2 x ) = ( x ( k 4 ) , x ( k + 1 ) ) . We must have 2 ( k 4 ) = k + 1 for ( x ( k 1 ) , x ( k + 1 ) ) to lie on our line. This gives k = 9
racodelitusmn

racodelitusmn

Expert

2022-07-08Added 5 answers

On the same line (!) of thought: the line l : y = 2 x is the same as the vector space Span { ( 1 , 2 ) } R 2 , or if you prefer: l : { ( r , 2 r ) / r R } , and then what we really want to happen is
( k 2 1 k k ) ( 1 2 ) = ( r 2 r ) k = r + 4 k = 2 r 1
so r + 4 = 2 r 1 r = 5 and thus k = 9

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