Recent content by HyperbolicMan

Thanks for the imput guys! I think I thought of an easy way to get my point across: Let x be a real number. What is the truth value of a statement such as "x=3" ? Based on what u guys have been saying, I'm going to guess that this is undefined. But it tackles the idea about a variable...

yes, but I think this only shows that (x_k+1, . . . , x_n) is basis for span(x_k+1, . . . , x_n), not for V/kerT.

Hi, I was working through this proof in my linear al textbook and there's this one step I can't get past. Any help would be appreciated. Homework Statement Let V be a finite dimensional vector space, and let T be a linear map defined on V. ker T \subseteq V and I am T \cong V/kerT Let...

Ahhh it seems like I should have been asking about universal quantification all along. If I'm following correctly, honestrosewater and alephzero, sets are defined by properties. Proving a statement for an arbitrary element of a set implies that the statement is true for all elements of the set...

The thought came to me while working with a function of the variable t. I was trying to figure out what its graph looked like. I was thinking about t varying over the real numbers and it occurred to me that this notion of 'varying' is kind of vague. So I thought about it, and I realized that...

For a circle, the derivative of the formula for area is the formula for circumference: d/dr([pi*r^2)=2*pi*r. Similarly, the derivative of the formula for the volume of a sphere is equal to the formula for surface area: d/dr(4/3*pi*r^3)=4*pi*r^2. I'm positive that these are consequences...

Just of out of curiosity, is it possible to rigorously define the notions of variable and constant? It seems to me that if these notions don't have rigorous definitions, then our way of thinking about mathematics pretty much falls apart.

Thanks for the help!

Let p be a continuous function such that for all r1, r2 in [0,R], ∫r2r1p(r)dr=(r22-r12)/R2. I'm trying to prove that p(r)=(2/R2)r. Question: Must p be unique? I'm not sure how to prove/disprove this.

Question: What is the probability that a random variable X with domain all real numbers will take a value in the closed interval [a,b]? It seems to me that in order to answer this question you have to know how the real numbers are distributed. Given the appropriate distribution function, you...

Finite Dimensional Inner-Product Space Equals its Dual!? Let V be a finite dimensional inner-product space. Then V is 'essentially' equal to its dual space V'. By the Reisz Representation theorem, V is isomorphic to V'. However, I've been told that V=V', which I am having a hard time...

"let x be a real number. Then x^2+1 does not equal 0." "For all x in lR, x^2+1 does not equal 0" As far as I know, both of these statements mean exactly the same thing. From a grammatical perspective, in the first statement, x is singular ("a real number"), while in the second, x appears...

What EXACTLY is a variable? This probably sounds like a really stupid question, buts been giving me a terrible headache . . . I've always had the intuitive understanding I learned in high school algebra that a variable can represent known or unknown quantity. I recently received an...