Linear Transformation and Diagonalization: Exercise Questions

LINEAR TRANSFORMATION AND DIAGONALIZATION

EXERCISE

1. For the following, prove that T is a linear transformation and find bases for both N (T) and R(T). Compute the nullity and rank of T and also determine whether T is one−to−one (1-1) or onto.

(a) T: R³→R² defined by T(a₁, a₂, a₃) = (a₁−a₂, 2a₃)

[Ans: Nullity = 1, rank =2, T is not one to one but it is onto].

(b) T: M_2×3 (F) → M_2×2 (F) defined by

 [Ans: Nullity = 4, rank = 2, T is neither one to one nor onto].

(c) T: P₂(R) → P₃(R) defined by T [f(x)] = xf(x) + f '(x).

[Ans: Nullity = 0, rank = 3, T is one−to−one but not on to].

2. Suppose that T: R² → R² is linear

T(1, 0) = (1, 4) and T(1, 1) = (2,5)

What is T (2, 3)?. Is T one−to−one?

[Ans: T (2, 3) = (5, 11). T is one−to−one]

3. 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)

4. Is there a linear transformation T: R³→R² such that T(1, 0, 3) = (1, 1) and T(−2, 0, −6)=(2, 1)? [Ans: No]

5. Let B and γ be the standard ordered bases for Rⁿ and R^m respectively. For each linear transformation

 T : Rⁿ → R^m compute [T]^γ_B

(a) T: R² → R³ defined by T(a₁, a₂) = (2a₁−a₂, 3a₁ +4α₂, a₁)

[Ans: ]

(b) T: R³→R defined by T(a₁, a₂, a₃)=2a₁+a₂−3a₃.

[Ans: [2 −1   3]]

(c) T: R³→R³ defined by

T (a₁, a₂, α₃) = (2a₂+ a₃, −a₁ + 4a₂ + 5a₃, a₁+a₃)

[Ans: ]

(d) T: Rⁿ→Rⁿ defined by

 T (a₁, a₂, a₃, ... a_n) = (a_n, a_n−1, a_{n −2},… a₁)

[Ans: ]

(e) T: Rⁿ→R defined by

T(a₁, a₂, a₃ ... a_n) = a₁ + a_n

[Ans: [1 0 0 0 1]]

6. Let T: R²→R³ be defined by

 T(a₁, a₂)= (a₁−a₂, a₁, 2a₁ + a₂). Let B be the standard ordered basis for R² and γ = {(1, 1, 0), (0, 1, 1), (2, 2, 3)), Compute [T]^γ_β. If α= {(1, 2), (2, 3)} compute [T]^β_α.

 [Ans: ]

7. Define T: M_2×2 (R) → P₂(R) by

= (a+b)+(2d)x + bx²

Let   and γ={1,x,x²} compute [T]^γ_β.

8. Let  B = (1,x,x²)

(a) Define T: M_2×2(F) → M_2×2(F) by T(A)=A^t. Compute [T]_α.

[Ans: ]

(b) Define T: P₂(R) → M_2×2 (R) by

  T[f(x)] = 

Where denotes differentiation. Compute [T]^γ_β        

[Ans:]

(c) If   then compute [A]_α.

 [Ans: ]

9. For the following matrices A ∈ M_n×n(F), find the eigenvalues, basis of eigenvectors, Q and D.

10. For each linear operator T on V, find the eigenvalues of T and an ordered basis B for V such that [T]_B is a diagonal matrix.

(a) V=R² and T (a, b) = (−2a+3b,− 10a +9b)

[Ans: λ=3, 4, B={ (3, 5), (1, 2) }}

(b) V=R³ and

T (a, b, c) = (7a−4b+ 10c, 4a−3b+8c,−2a+b−2c)

[Ans: λ=−1, 1, 2, B={(1, 2, 0), (1,−1,−1), (2, 0, −1)}

(c) V=P₃(R) and T[f(x)] = f(x) +ƒ (2) x

[Ans: λ=1,3, B={−2+x, −4+x², −8+x³, x}

11. For each of the following matrices A ∈ M_n×n(R) test A for diagonalizability and if A is diagonalizable, find an invertible matrix Q and a diagonal matrix D such that Q^‒1AQ= D.

12. For each of the following linear operators T on a vector space V, test T for diagonalizability and if T is diagonalizable, find a basis. B for V such that [T]_B is a diagonal matrix.

(a) V=P₃ (R) and T is defined by

T[f(x)]=ƒ'(x) + f "(x) respectively.

[Ans: Not diagonalizable]

(b) V=R³ and T is defined by

 [Ans: Not diagonalizable]

 (c) V=P₂(R) and T is defined by

 T [ƒ (x)] = ƒ(0) + ƒ(1)(x + x²)

 [Ans: B={x−x², 1−x−x²,x²+x}]

(d) V=C² and T is defined by

 T (z, w) = (z+ iw, iz+w)

[Ans: B={(1, 1), (1, 1) }]

13. Find the general solution for the following system of differential equations.

(a) x'₁ = 8x₁+10x²

x'₂ = −5x₁‒7x²

[Ans: ]