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A, B and C are on a betting game. B loses Php 350 of his money to A. As a result, A now has twice as much as what is left with B. Then, C loses Php 700 to B. As a consequence, C now has only one-third as much money as B would then have. If A has loses Php 210 to C, C will have as much as money as A would have left. How much did each have at the start?
Consider A = <0,4,2> , B = <6,-1,0> , and C = <3,0,1>. Find scalars a, b and c such that aA + bB = (c - 1)C.
.
1.1 Prove that the characteristic polynomial of a 2 × 2 matrix A can be expressed as
λ^2 − tr(A)λ + det(A), where tr(A) is the trace of A.
In Exercises 15–16, determine whether the expression makes sense mathematically. If not, explain why?
15. (a) u · (v · w)
(b) u · (v + w)
(c) u · v
(d) (u · v) − u
Suppose that a vector a in the xy-plane points in a direction that is 47◦ counterclockwise from the positive x-axis, and a vector b in that plane points in a direction that is 43◦ clockwise from the positive x-axis. What can you say about the value of a · b?
Define W = x y : xy ≥ 0 . Decide if V is a vector space or notand prove your claim. (Hint: V is the union of the first and third quadrants in the xyplane)
Define W =
x
y
: xy ≥ 0
. Decide if V is a vector space or notand prove
your claim. (Hint: V is the union of the first and third quadrants in the xyplane)
. Define V =
x
y
: x, y ≥ 0
. Decide if V is a vector space or not and prove
your claim. (Hint: V is the first quadrant in the xy-plane).
Which of the following is a linear equation in x; y and z?
1. −x−1 + e−
√
2y = 3z, where e = 2.71828 ....
2. 2π ln(e−1
z ) − 2y + z = ln(3) − x.
3.
p
y2 + 4y − 2z = 7x.
4. y + 4y − 2z = 7x−2.
QUESTION 2 2.1. Find the change of basis matrix P∁←ℬ for the bases
ℬ = {(9, 2), (4, −3)} and ∁= {(2, 1), (−3, 1)} of ℝ2 .
2.2.Verify [v]∁ = P∁←ℬ[v]ℬ for v = (−5, 3).
