I need help for all 1 to 5 problems. The question is in the attachment. Please be as specific as and as clear as possible. The text book I used is Linear Algebra Done right and the relevant chapters are 5 and 6.

R3 is R3

1. Let T ∈ L(R 3 ) be given by

T(a, b, c) = (b + c, 2b, −2a + b + 3c).

You can assume as given the fact that the eigenvalues of T are 1 and 2.

(a) Find a basis of the eigenspace for each eigenvalue of T.

(b) Determine whether T is diagonalizable.

2. Let R3 be equipped with the standard inner-product (i.e., the dot-product).

Find an orthogonal basis {v1, v2, v3} of R3 such that v1 = (1, 2, 3).

3. Let U be the plane in R3 given by 2x − y − 2z = 0. Compute the shortest

distance from (1, 1, 2) to a point in U.

4. Let T1 : R3 → R2 and T2 : R2 → R3 be linear maps. (a) Prove that the linear

operator T2T1 : R3 → R3 is not invertible.

(a) Prove that the linear operator T2T1 : R3 → R3 is not invertible.

(b) Is T1T2 : R2 → R2 necessarily not invertible? Either prove this or give an

example of T1 and T2 such that T1T2 is invertible.

5.

Suppose V is finite-dimensional, T ∈ L(V ), and U is a subspace of V . Prove

that U and U ⊥ are both invariant under T if and only if P u T = T Pu .