T is Linear

(a) T is linear

Problem 1

Solution :

Problem 2

Solution :

Here, (1) ≠ (2) => Condition (b) is not satisfied.

So, T is not linear.

Problem 3.

Let  Prove that T is not linear.

Solution :

Here, (1) ≠ (2) => Condition (b) is not satisfied.

So, T is not linear.

Problem 4.

Let  Show that T is not linear.

Solution :

Here, (1) ≠ (2) => Condition (b) is not satisfied.

So, T is not linear.

Problem 5.

Prove that there exists a linear transformation T : R² → R³ such that T(1, 1) = (1, 0, 2) and T(2, 3) = (1, -1, 4). What is T(8, 11)?

Solution :

Given: The transformation T : R² → R³ is defined by,

T(1, 1) = (1, 0, 2) and T(2, 3) = (1, -1, 4)

To prove that the transformation T is linear, and then find the value of T(8, 11).

First, prove that if the set S = {(1, 1), (2, 3)} is linearly independent.

Whenever a(1, 1) + b(2, 3) = (0, 0), the result would be a = 0, b = 0

=> S = {(1, 1), (2, 3)} is linearly independent.

Furthermore, the dimension of R² = 2 = number of vectors in S.

=> S spans R².

Therefore, S is the basis for R².

So, every vector in the domain R² can be written as a linear combination of these vectors.

Therefore, it proves that the transformation T is linear.

Let (a, b) be any vector in R². Then,

Take T on both sides, we get

Problem 6.

Is there a linear transformation T : R³ → R² such that T(1, 0, 3) = (1, 1) and T(-2, 0, -6) = (2, 1)?

Solution :

We know that,

If T is linear, then

(a) T(x + y) = T(x) + T(y)

(b) T(cx) = c T(x)

Given:

So, T is not linear.

Problem 7.

Prove that there is an additive function

i.e., T(x,y) = T(x) + T(y). T : R → R that is not linear.

Solution :

Let T : R → R, Suppose x, y Є R are distinct vectors.

Define by T(v) = f(v) for all v Є V. So that, T is an additive.

Prove that T is not linear.

Hence, T is not linear.

Problem 8.

Let 

Prove that T is linear.

Solution :

Verify T(ax+y) = a T(x) + T(y)

Then

And

From (2) & (3) we get

T(ax + y) = a T(x) + T(y) To

Hence, the transformation T: R³ → R²; defined by  is linear.

Problem 9.

Let T: R² → R³ defined by  Prove that T is linear.

Solution :

Problem 10

Prove that T is linear

Solution :

Also

Problem 11

Prove that T is linear

Solution :

Given that  defined by 

To prove T is linear, we consider

First part of the definition is satisfied.

Problem 12.

Solution :

Let the transformation T : M_{n x n}(F) → F defined by

To prove that T(A) = tr(A) is linear transformation.

Thus, T(c P) = c T(P)

=> T(A) = tr(A) satisfies the properties that