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

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 ...

Solution Summary

Solutions to the posted problem given with step by step working so that the student could easily understand and use the solutions to solve other similar problems.

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