Linear Algebra Answers

Define: R^3→R^3 by

T(x,y,z)=(x-y+z,x+y,y+z)

Let v1= (1,1,1), v2= (0,1,1), v3= (0,0,1). Find a matrix of T with respect to the basis {v1,v2,v3}. Futher check T is invertible or not.

Complete {(2,0,3)} to form an orthogonal basis of R^3

Consider the basis S = {v₁, v₂} for R², where v₁ = (− 2, 1) and v₂ = (1, 3), and let T:R² → R³ be the linear transformation such that

T(v₁) = (− 1, 4, 0) and T(v₂) = (0, − 5, 9)

Find a formula for T(x₁, x₂), and use that formula to find T(4, − 5).

Give exact answers in the form of a fraction.

Suppose that a 4 × 4 matrix A has eigenvalues λ₁ = 1, λ₂ = − 4, λ₃ = 6, and λ₄ = − 6. Use the following method to find det (A).

If

p(λ) = det (λI − A) = λⁿ + c₁λ^{n − 1} + ⋯ + c_n

So, on setting λ = 0, we obtain that

det (− A) = c_n or det (A) = (− 1)ⁿc_n

det (A) = ?

Apply the fundamental theorem of 

homomorphism to prove that :

R⁴/R² isomorphic to R²

Let T : R2-+ R2be a linear operator with 

matrix [ (w.r.t. the standard basis).

[ 7 1

-1 -1]

Use Cayley-Hamilton theorem to 

check whether T is invertible or not. If T is 

invertible, obtain T-1(x, y) for 

(x,y) E R2 . If T is not invertible, obtain 

the minimum polynomial of T.

Which of the following statements are true ? 

Give reasons for your answers in the form of a 

short proof or a counter-example : 10 

(1) If X={D|D is a 2 x 2 diagonal matrix) and 

D1~D2 iff D1 = cD2, c not equal to 0, c E Z' , then~ is an equivalence relation on X. 

(ii) If two matrices A and B have the same 

eigen values, then A = B. 

(iii) Any symmetric matrix is non-singular. 

(iv) If V = {(x, y) belongs to R² |x = y} , then (1, 0) +V and (0, 1) + V are two distinct elements of 

R2/V.

Let : 

V = R3, 

W = {(x1, X2, X3) ! x1 - x2 = .X3}. 

Show that W is a subspace of V. Further, 

find a basis for W, and hence, find the 

dimension of W.

The function f₁ and f₂ are linearly independent if there is a real number X such that k₁f₁(x) + k₂f₂(x)=0

(2) Prove that if V is a subspace of a vector space W and if V is infinite demensial then so in W.

(3) Prove that if A is digonalizble matrix then the rank of A is the number of non-zero eigen values of A.

Solve by finding the basis over R for the solution space.

(A) X + 3y -3z=0

2x - 3y + z=0

3x -2y + 2z=0

(B) X + Y + Z + W=0

2X + 3Y - Z +W=0

3X + 4Y +2W=0