Define linear transformation with an example. Is a transformation defined by T(x, y) = (3x + y, 5x + 7y, x + 3y) linear? Justify.

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i) Solution:

T(x, y) = (3x + y, 5x + 2y, x + 3y)

Let u = (x1, y1)

v = (x2, y2)

Now,

T(\(\vec{u} + \vec{v}\)) = T(x1 + x2, y1 + y2)

= (3(x1 + x2) + y1 + y2), 5(x1 + x2) + 2(y1 + y2), (x1 + x2) + 3(y1 + y2))

∴ T(u + v) = (3×1 + 3×2 + y1 + y2, 5×1 + 5×2 + 2y1 + 2y2, x1 + x3 + 3y1 + 3y2,)

Now,

\(T(\vec{u}) + T(\vec{v})\) = T(x1, y1) + T(x2, y2)

= (3×1 + y1, 5×1 + 2y1, x1 + 3y1) + (3×2 + y2, 5×2 + 2y2, x2 + 3y2)

= (3×1 + 3×2 + y1 + y2, 5×1 + 2y1 + 5×2, 2y2, x1 + x2 + 3y1 + 3y2)

Since, T(\(\vec{u} + \vec{v}\)) = \(T(\vec{u}) + T(\vec{v})\)

So, it is linear transformation

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