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Linear Operator - Basis -Kernel-Range-Linear Transformation.

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Question (1):
Let T: R^3 into R^3 be a linear transformation defined by

T(x, y, z) = (x + 2y - z, y + z, x + y - 2z )

Find a basis and the dimension of ( i ) Range of T ( ii) the Kernel of T

Question (2) :

If T:R^4 into R^3 is a linear transformation defined by

T( a, b, c, d) = ( a - b + c + d, a + 2c - d, a + b + 3c - 3d ) for a, b, c, d belong to R, then find a basis for Ker T ( i.e. Null Space of T) and Range of T.

For the description of the questions with symbolic usage, please see the attached question file.

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Solution (1)
T(x, y, z) = (x + 2y - z, y + z, x + y - 2z )
Let alpha = (x, y, z). Now for the basis of Ker T ...........

T(alpha) = T(x, y, z) = (x + 2y - z, y + z, x + y - 2z ) = (0, 0, 0 )
The associated matrix of coefficients .................
........................
...............................
It is in ...

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