Question

# let let T : R^3 --> R^2 be a linear transformation defined by T ( x,...

let let T : R^3 --> R^2 be a linear transformation defined by T ( x, y , z) = ( x-2y -z , 2x + 4y - 2z) a give an example of two elements in K ev( T ) and show that these sum i also an element of K er( T)

Solution:

The given linear transformation is defined by T( x, y, z) = ( x - 2y - z , 2x + 4y - 2z). Now let ( x, y, z ) and ( a, b, c ) be two elements in Ker(T). Then we have T(x, y, z ) = 0 and T(a, b, c) = 0 i.e.   ( x - 2y - z , 2x + 4y - 2z) = 0 and ( a - 2b - c , 2a + 4b - 2c) = 0 .

Now, T(x+a, y+b, z+c) = ( x+a - 2y - 2b - z - c , 2x +2a + 4y + 4b - 2z - 2c)

=  ( x - 2y - z , 2x + 4y - 2z) + ( a - 2b - c , 2a + 4b - 2c)

= 0 + 0 = 0.

This show that (x+a, y+b, z+c) is in Ker(T)   i.e. (x, y, z ) + (a, b, c ) is in Ker(T) .

This completess the proof.

RESULT: Since Ker(T) is a subspace of , so sum of any two elements of Ker(T) is in Ker(T). The proof of this result is given above.

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