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MATH2501 Linear Algebra, S1 2015: Problems

1. LINEAR EQUATIONS AND MATRICES

1. For each of the following matrices A and vectors b, use Gaussian elimination to find the general

solution of the system Ax = b.

8. Let u, v, w be elements of a vector space V , and suppose that the set S = {u, v } is linearly

independent. Prove that {u, v, w } is linearly dependent if and only if w is in span(S).

9. Some linear independence problems using more advanced techniques.

a) Show that { cos t, sin t, t cos t, t sin t } is independent by considering a possible identity

λ1 cos t + λ2 sin t + λ3t cos t + λ4t sin t = 0 and evaluating the left hand side at carefully

chosen values of t.

b) Use differentiation to show that the set { 1, ex, e2x } is linearly independent.

c) Let f(t) = λ1 cos t+ λ2 sin t+ λ3 cos 2t+ λ4 sin 2t. By evaluating

show that the set of functions { cos t, sin t, cos 2t, sin 2t } is linearly independent.

10. Let V be the vector space of all twice differentiable real–valued functions defined on R. If f, g

are in V then the determinant

Wf,g(t) =

∣∣∣∣f(t) g(t)f ′(t) g′(t)

∣∣∣∣ = f(t)g′(t)− f ′(t)g(t)

is called the Wronskian of f and g; it is significant in the theory of linear ordinary differential

equations.

a) Prove that if { f, g } is a linearly dependent set in V then Wf,g(t) = 0 for all t.

b) Show that { cos t, sin t } is a linearly independent set.

11. Let S = { (1,−1, 3), (−1, 3,−7), (2, 1, 0) }. Do the vectors u = (5, 1, 3) and v = (2, 3, 6) belong

to span(S)?

12. Let S = { 1 − t + 3t2, 1 − t2, 2 + t + 5t2 }. Does the polynomial p(t) = 1 + t + t2 belong to

span(S)?

13. Let S = { 1− t+ 3t2, 1− t2, 2 + t+ 5t2 }. Does the polynomial p(t) = 12941− 7696t+50114t2

belong to span(S)? Suggestion. Look back at the previous problem before you do any annoying

calculations.

14. In the following questions, try to do as little calculation as possible. Give reasons for all your

answers.

3. Let T : R2 → R2 be a linear transformation. Find a formula for T (x1, x2), given that

a) T (1, 0) = (3, 4) and T (0, 1) = (4, 9);

b) T (4, 7) = (3, 4) and T (3, 5) = (4, 9);

c) T (5, 7) = (3, 4) and T (2, 7) = (2, 5).

4. For each linear transformation in question 2 having finite–dimensional domain and codomain,

find the matrix of the transformation with respect to standard bases.

7. Given that the linear function T : R2 → R2 has matrix A with respect to the standard basis of

with respect to the basis B = { (1, 3), (3, 7) } of R2, find the matrix of T with respect to the

standard basis.

9. Let v1 = (1,−2, 0), v2 = (0,−1, 1), v3 = (1, 0,−1); suppose that T : R

3 → R3 is a linear

transformation and that

T (v1) = 5v1 + v2 , T (v2) = 5v2 + v3 , T (v3) = 5v3 .

a) Write down the matrix of T with respect to the basis consisting of the vectors v3, v2, v1,

in that order.

b) Find the matrix of T with respect to the standard basis for R3. Comment. This kind of

problem will be exceedingly important in Chapter 10.

10. Find bases for the kernels and images of the following linear transformations. Hence obtain the

nullity and rank of each transformation.

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