Ok, so assuming we reduce the set in U to a basis: then if v is in U the given generating set for W is linearly dependent. Then v would be removed from this set, making it linearly independent. This implies that U and V share a basis, and thus their dimensions are the same. If v is not in U...
Homework Statement
Given v1, v2 ... vk and v, let U = span {v1, v2 ... vk} and W = span {v1, v2 ... vk, v}. Show that either dim W = dim U or dim W = 1 + dim U.
The Attempt at a Solution
I'm not really sure where to start. If I knew that {v1, v2 ... vk} was linearly independent, then it would...
Thanks for the reply! Today I was able to prove (1) by contraposition but I thought that meant I had to prove (2) by contraposition as well. Can I just prove the positive statement for (2)?
Edit: Wait, taking statement A as {p, q, pq} is linearly independent and statement B as deg p ≥ 1 and deg...
Homework Statement
Let {p, q} be linearly independent polynomials. Show that {p, q, pq} is linearly independent if and only if deg p ≥ 1 and deg q ≥ 1.
Homework Equations
λ1p + λ2q = 0 ⇔ λ1 = λ2 = 0
The Attempt at a Solution
λ1p + λ2q + λ3pq = 0
I know if λ3 = 0, then the coefficients of...
Homework Statement
Is U = {A| A \in nℝn, A is invertible} a subspace of nℝn, the space of all nxn matrices?
The Attempt at a Solution
This is easy to prove if you assume the regular operations of vector addition and scalar multiplication. Then the Identity matrix is in the set but 0*I and...
Do you know what I did? I simplified the original function to get the one I posted here, just checked and they're not the same. That's why the derivative didn't match up. :surprised I think I've about got it sorted out now.
Somebody want to help with this derivative:
y = (7.75x3 + 2250) / 750x
I differentiate it with the quotient rule and get:
dy/dx = 31x / 1500 - 3 / x2
But that's wrong. It's got a zero around 5 - 6 and the original function has its minimum at (4.03, 1.12). Not sure what I did wrong.
No, I multiplied that in at the very beginning. I scaled up the fraction by 1000 times so I had a 4 in the numerator rather than a decimal. After fixing what Reptilian mentioned, my derivative looks correct, and the zero is at the x-value of that local minimum, so I think it's right.
Using the quotient rule on the 4∏r2 / 3000 term. The denominator of the differentiated expression is 16∏2r4. A pi and an r cancel from the numerator which was 8∏r(3000).
Yeah I noticed. No worries. :tongue: Do you know what's wrong with the derivative. I found just with the power rule and the quotient rule on the last term.
After simplifying I get:
(0.003 + 0.004√2) ∏r2 + 3/r - (4∏r2) / 300
Which is equivalent now according to a graphing calculator.
Differentiating, I get:
(0.006 + 0.008√2) ∏r - 3r-2 - 1500/∏r3
But thats not right. i.e. it has positive values when the slope of the function is negative. :/
Does somebody want to help me with differentiating this function:
(0.003 + 0.004\sqrt{2})∏r2 + 0.004∏r [(2250 - ∏r3) / 3∏r2)]
Originally I tried simplifying it by distributing the 0.004∏r into the fraction, but whenever I did that the two expressions were never equivalent when I checked them...