5) Please help with this urgently

Question

Let T : R² → R² be a linear operator defined by

T ⌈1⌉ = ⌈2⌉ T ⌈2⌉ = ⌈4⌉

⌊1⌋ ⌊2⌋ , ⌊1⌋ ⌊5⌋

Find a formula for T ⌈x⌉

⌊y⌋

Answer 1

If I understood your question correctly, we need to find a linear operator T: ℝ² → ℝ² such that T(1,1) = (2,2) and T(2,1) = (4,5).

Since T is a linear operator, it has the form:

1. T(x,y) = (ax + by, cx + dy) for some constants a, b, c, and d.

Finding the constants:

Using the condition T(1,1) = (2,2):

1. a(1) + b(1) = 2 → a + b = 2
2. c(1) + d(1) = 2 → c + d = 2

Using the condition T(2,1) = (4,5):

1. a(2) + b(1) = 4 → 2a + b = 4
2. c(2) + d(1) = 5 → 2c + d = 5

Solving for a and b: From 2a + b = 4 and a + b = 2, subtract the second from the first: a = 2

Substitute back: 2 + b = 2, so b = 0

Solving for c and d: From 2c + d = 5 and c + d = 2, subtract the second from the first: c = 3

Substituting back we get: 3 + d = 2, so d = -1

Answer:

T(x,y) = (2x, 3x - y)

Verification:

1. T(1,1) = (2·1, 3·1 - 1) = (2, 2) ✓
2. T(2,1) = (2·2, 3·2 - 1) = (4, 5) ✓

Sorry its a bit late but maybe it should have been tagged as Linear Algebra. Hope you did well in your class. Cheers!