Which of the following transformations are linear y1 2x1 y2

Which of the following transformations are linear? {y_1 = 2x_1 y_2 = 5 {y_1 = 5 y_2 = 7 {y_1 = 4 y_2 = 9 {y_1 = 7x_1 + x_2 y_2 = -x_1 {y_1 = 0 y_2 = 4x_2 {y_1 = 5x_1 - 8x_2 + 10x_3 y_2 = 6x_2 - 7x_3 y_3 = -9x_1 - 4x_2 {y_1 = x_2^2 y_2 = x_3 y_3 = x_1 You have attempted this problem 0 times. You have 2 attempts remaining.

Solution

The transformations which are linear are: C, D, E

They are linear transformation because they satisfy the properties of linearity, i.e. a transformation T:V V is said to be linear if T(u+v) = T(u) + T(v) for all u, v V and T(cu) = cT(u) for all c R, u V

Also T(0) = 0 (i.e. T sends zero vector to the zero vector)

The reason that A, B for not being linear transformation is the presence of non-zero constant term.

That is for option A, y₂ = 5 and for option B, y₁ = 7, y₂ = 4, y₃ = 9.

Option F is non-linear due to the presence of the squared term x₂²