Homework answers / question archive / 1) Let S and T be the mappings of R2 into R2 defined by S(x1, x2) = (x1 + x2, x1 – x2) and T(x1, x2) = (-x2,-x1)

1) Let S and T be the mappings of R2 into R2 defined by S(x1, x2) = (x1 + x2, x1 – x2) and T(x1, x2) = (-x2,-x1)

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1) Let S and T be the mappings of R² into R² defined by S(x₁, x₂) = (x₁ + x₂, x₁ – x₂) and T(x₁, x₂) = (-x₂,-x₁).

(a) Prove that each of S and T is a linear operator.

(b) Find S + T and 2S – 3T.

2. Determine whether the given mapping T: R² ® R² is a linear operator.

(a) T(x, y) = (x – y, 0)         (b) T(x, y) = (xy, x)                (c) T(x, y) = (x + 1, y – 1)

(d) T(x, y) = (x + y, x – y)        (e) T(x, y) = x(2, 1)            (f) T(x, y) = (x – y)(x + y, 0)

3. Determine whether the given mapping T : R³ ® R² is a linear transformation.

(a) T(x, y, z) = (2x + y, x + z)         (b) T(x, y, z) = (x – y, x² – y²)

(c) T(x, y, z) = (x + y + 1, x + y -1)                (d) T(x, y, z) = (x + y + z, 0)

4. Determine which of the following mappings T : p₁ ® p₂ are linear operators.

(a) T(a₀ + a₁x) = a₀x                     (b) T(a₀ + a₁x) = a₀ + a₁(x + 1)

(c) T(a₀ + a₁x) = a₀a₁ + a₀x            (d) T(a₀ + a₁x) = a₀ + a₁ +a₀x