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The T: R 4 → R 4 , given by T (x, y, z, w) =...

The T: R 4 → R 4 , given by T (x, y, z, w) = (x + y, y, z, 2z + 1) is a linear transformation? Justify that.

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Answer #1

Let (x1, y1, z1, w1) ∈ R4 and (x2, y2, z2, w2) ∈ R4​​​​​​

This implies that, (x1 + x2, y1 + y2, z1 + z2, w1 + w2) ∈ R4

.

Now we have,

T(x1, y1, z1, w1) = (x1 + y1, y1, z1, 2z1 + 1)

and, T(x2, y2, z2, w2) = (x2 + y2, y2, z2, 2z2 + 1)

.

Now,

T(x1 + x2, y1 + y2, z1 + z2, w1 + w2)

= (x1 + x2 + y1 + y2, y1 + y2, z1 + z2, 2z1 + 2z2 + 1)

= (x1 + y1, y1, z1, 2z1 + 1) + (x2 + y2, y2, z2, 2z2)

= T(x1, y1, z1, w1) + (x2 + y2, y2, z2, 2z2)

≠ T(x1, y1, z1, w1) + T(x2, y2, z2, w2)

.

This proves that T is not linear.

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