MATHS 120: ALGEBRA SEMESTER TWO 2021
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SEMESTER TWO 2021
MATHS 120: ALGEBRA
1. Let p1 = '(┌)1_11
, p2 = '(┌)
and p3 = '(┌)13_2
. (a) Show that p1 , p2 and p3 are not collinear.
(b) Find a Cartesian equation for the plane P containing p1 , p2 and p3 .
(c) Note that the coordinates of p1 , p2 and p3 are all integers,
and the second coordinates are _1, 1 and 3, which are all odd integers. Is there an element of the plane P such that its coordinates are all integers and its second coordinate is even ?
2. Let z = 3 _ 4i e C. (a) What is |z2 |?
(b) For which n e N is “|zn | > 10n” true? (Hint: use induction.)
3. Let
f : R _- R
┌x┐ |- ┌x + 1┐
and
g : R2 _- R
┌ ┐y(x) |- ┌ _1┐ :
(a) Which of f , g , f · g and g · f are well-defined functions?
(b) Of the ones that are well-defined, which are linear transformations?
4. Let u1 = ┌ _21┐ , u2 = ┌ 1_1┐ , v1 = ┌ ┐2(1) , v2 = ┌ ┐3(3) , and v3 = ┌ ┐0(0) . Let U = {u1 ; u2 } and let V = {v1 ; v2 ; v3 }.
(a) Let c1 u1 + c2 u2 = ┌ ┐y(x) : Solve for c1 and c2 in terms of x and y .
(b) Is U a basis for R2 ?
(c) Is V a basis for R2 ?
(d) Let L : R2 - R2 be the unique linear transformation such that L(u1 ) = v1 and L(u2 ) = v2 . What is the matrix associated with L (with respect to the standard basis)?
(e) How many linear transformations F : R2 - R2 are there such that F (u1 ); F (u2 ) e V?
5. Let n 2 1, let A and B be n × n matrices, let X = AB and Y = B(In + A)t _ B . [8 marks]
Which of the following statements are true?
(a) X = Y .
(b) det(X) = det(Y).
2022-10-19