EP 307: Quantum Mechanics ILinear Algebra Tutorial Sheet

This tutorial sheet deals with some problems aimed at refereshing your knowledge oflinear algebra.

1. Consider the vector space P 3(x) spanned by polynomials of degree three. Using thestandard basis {1, x, x2, x3}, obtain a representation for the �rst-derivative operatorddx

in P3.

2. Show that the vectors

α1 = (1, 0,−1), α2 = (1, 2, 1), , α3 = (0,−3, 2)

form a basis for R3. Express the coordinates of the standard basis vectors in the newbasis set.

3. Let B = {α1, α2, α3} be the ordered basis for R3 consisting of

α1 = (1, 0,−1), α2 = (1, 1, 1), α3 = (1, 0, 0).

What are the coordinates of the vector (a, b, c) in the ordered basis B.

4. Let T : R3 → R3, be a linear transformation de�ned by

T (x1, x2, x3) = (2x1 + x2, x1 + x2 + 3x3,−x2).

(a) Obtain the representation of T with respect to the standard basis.

(b) Let there be another basis B = (v1, v2, v3), where v1 = (−1, 0, 0), v2 = (2, 1, 0),v3 = (1, 1, 1). Obtain the representation of T with respect to B, by applying asuitable similarity transformation on the result of part (a).

5. Consider the matrix

A =

3 0 20 2 0−2 −0 −1

.

Is A a diagonalizable matrix?. If yes, demonstrate it by �nding a similarity transfor-mation P which will diagonalize it. If no, give reasons.

6. Let A be a 2× 2 matrix with real entries. For X, Y in R2×1 let

fA(X, Y ) = Y TAX.

Show that fA is an inner product on R2×1 if and only if A = AT , A11 > 0, A22 > 0,and det (A) > 0.

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7. Consider the space of polynomials with the degree at most two (P 2(x)). Assume thatthe de�nition of the inner product on this space is

(p|q) =∫ 1

0p(x)q(x)dx,

where p(x), q(x) ∈ P 2(x).

(a) Clearly the standard basis set for P 2(x) is B = {1, x, x2}. Is this basis setorthogonal?

(b) If your answer to part (a) is no, then obtain an orthonormal basis set for P 2(x)using the Gram-Schmidt procedure.

8. Consider the real-symmetric matrix

A =

3 −2 0−2 3 0

0 0 5

.

Obtain its eigenvalues and eigenvectors, and thus, construct an orthogonal matrix Owhich will diagonalize A.

9. Consider the matrix

A =

2 −1 1−1 2 −1

1 −1 2

Calculate eA. Is e−A inverse of eA? Support your answer by calculations.

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