No they won’t be.Consider the linear combination (1)(u – v) + (1) (v – w) + (-1)(u – w).This will add to 0. But the coefficients aren’t all 0.Therefore, those vectors aren’t linearly independent.
You can try an example of this with (1, 0, 0), (0, 1, 0), and (0, 0, 1), the usual basis vectors of R3.
That method relied on spotting the solution immediately.If you couldn’t see that, then there’s another approach to the problem.
We know that u, v, w are linearly independent vectors.So if au + bv + cw = 0, then a, b, and c are all 0 by definition.
Suppose we wanted to ask whether u – v, v – w, and u – w are linearly independent.Then we’d like to see if there are non-zero coefficients in the linear combinationd(u – v) + e(v – w) + f(u – w) = 0, where d, e, and f are scalars.
Distributing, we get du – dv + ev – ew + fu – fw = 0.Then regrouping by vector: (d + f)u + (-d +e)v + (-e – f)w = 0.
But now we have a linear combo of u, v, and w vectors.Therefore, all the coefficients must be 0.So d + f = 0, -d + e = 0, and –e – f = 0. It turns out that there’s a free variable in this solution.Say you let d be the free variable.Then we see f = -d and e = d.
Then any solution of the form (d, e, f) = (d, d, -d) will make (d + f)u + (-d +e)v + (-e – f)w = 0 a true statement.
Let d = 1 and you get our original solution. You can let d = 2, 3, or anything if you want.
Answer:
The length of the third side is ( x - 2 )
Step-by-step explanation:
Volume of a box is the product of its length, width and height.
Volume of box = length × width × height.
One side is (x - 4) and another side is (2x + 1).
These given sides can be either length, height or width.
The volume of a box is given by V = 2x3-11x2 +10x +8. This means each of the sides are factors of the volume. To get the third side, we divide the volume by the product of the two given sides.
Product of the two given sides =
( x-4 ) (2x + 1) = 2x^2 + x - 8x + 4
= 2x^2 - 7x - 4
The long division is shown on the attached photo
The length of the third side is ( x - 2 )
Answer:
1. 5 / 6
2. 1 / 3
3. 3 / 6 or 1 / 2
Step-by-step explanation:
The probability that A is not yellow is 5 / 6
The probability that B is blue is 1 / 3
The probability that A is red is 3 / 6 or 1 / 2
The relationship is not proportional because the boys and girls are in different grades and they are in different activities.
