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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Suresh Chand · Accepted Answer

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) = (3x1 + 3x2 + y1 + y2, 5x1 + 5x2 + 2y1 + 2y2, x1 + x3 + 3y1 + 3y2,) Now, (T(vec{u}) + T(vec{v})) = T(x1, y1) + T(x2, y2) = (3x1 + y1, 5x1 + 2y1, x1 + 3y1) + (3x2 + y2, 5x2 + 2y2, x2 + 3y2) = (3x1 + 3x2 + y1 + y2, 5x1 + 2y1 + 5x2, 2y2, x1 + x2 + 3y1 + 3y2) Since, T((vec{u} + vec{v})) = (T(vec{u}) + T(vec{v})) So, it is linear transformation

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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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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