Linear Algebra Answers

Determine whether T is linear transformation or not.

T:R2−→R3

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

If p2(x) = ax2 + bx + c be a polynomial and T : p2 −→ p2 be a linear transformation given by T(p) = p(2x+1). For the basis for p2 as u = (1,x,x2). Is T diagonalizable, if yes diagonalize this transformation.

Consider the vector space V = C²with scalar multiplication over the real numbers R, and let T : V → V be the linear operator defined by T (z1, z2) = (z1 − iz2, z2 − z2) . Let W be the cyclic subspace of V generated by w = (1 + 2i, 1 + i).

5.1 Find the T–cyclic basis β for W generated by w.

5.2 Find the characteristic polynomial of TW .

5.3 Find [TW ]β. 5.4 Explain whether T = TW

Let T : V → V be a linear operator on a finite-dimensional vector space V over C such that T² = T.

7.1 Show that R(T) ⊆ N(T − I) and R(T − I) ⊆ N(T).

7.2 Show that V = R(T) + R(T − I).

7.3 Show that V = N(T) ⊕ N(T − I).

7.4 Show that T is diagonalisable

A square matrix is nonsingular if it can be written as a product of elementary matrices.

Consider a linear mapping f : V→W with dim V = n and dim W = m. Prove that

i) Nullity(f) = 0 if f is one-to-one

ii) f is onto if R(f) = m

Let T: P₂→P₂ be the mapping defined by

T(a₀ + a₁x + a₂x²) = 3a₀ + a₁x + (a₀ + a₁)x²

i) Show that T is linear

ii) Find a basis for the kernel of T

iii) Find a basis for the range of T