Instructions: Attempt all problems. You should provide appropriate justification for you
answers and refrain from using formulac
esults that have not been discussed in lectures.
Unsubstantiated answers will not receive full credit.
1. [10] Find, with justification, a homogencous system of lincar equations in fou
variables whose solution sct is equal to
2 1 5 5
So —1 0 -3 —2
SPA ge || 2 || 3
1 3 0 5
(Hint: Take a generic vector # € RY. When is # a lincar combination of the fou
given vectors?).
2. [12] A square matrix A is said to be idempotent if A2 = A.
(a) Find all idempotent 2 x 2 matrices whose second column is the zero vector.
Show your work.
(b) By a result from lectures, the mapping
L: R25 R? 7s proj;
is linear. Find the standard matrix of L, and verify that it is idempotent.
(c) Is it true that for any non-zero vector @ € R”, the standard matrix of the linca
mapping
L:R" = R", Zw proj; ¥
is idempotent? Give a
ief explanation of your answer.
3. [10] In cach of the following cases, determine (with justification) whether the given
mapping L is lincar. In the cases where it is, find the standard matrix of L.
G LR SR, | 7d" Ware, 20
zy if mma <0
(ii) L: R? — R%, [ mn ] + x XXXXXXXXXX, where i and arc fixed (but unknown)
Ta
vectors in R2.
4. [10] There is a unique lincar mapping L: R* — R? such that
1 —1 1
D1] (2D 12) wee] D120)
0 1 - —1
Find, with justification, the standard matrix of L.
5. [8] The matrix
2 21
XXXXXXXXXX
3 1 22
is the standard matrix of the lincar mapping L: R* — R? given by reflection about
a plane P in 3-space that passes through the origin. Find a scalar equation for P
(Hint: What docs L do to the vectors on P?).