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  1. Let $F$ be a constant unit force that is parallel to the vector $(1, 0, 1)$ in xyz-space. What is the work done by $F$ on a particle that moves along the path given by $(t, t^2, t^3)$ between time $t=0$ and time $t=1$?

  2. Let $f$ be real valued function defined and continuous on the set of real numbers $R$. Is it true that the set $S=\{f(c): 0<c<1\}$ is a bounded subset of $R$?

  3. Let $V$ be the real vector space of all real $2\times 3$ matrices and let $W$ be the real vector space of all real $4\times 1$ column vectors. If $T$ is a linear transformation from $V$ onto $W$. What is the dimension of the subspace $\{v\in V: T(v)=0\}$?

Thanks in advance.

Mikasa
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Fischer
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  • 1.) $F$ is conservative (find a potential function $\phi$), the line integral only depends on the end points of the path. – BobaFret Oct 27 '12 at 01:15

1 Answers1

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Hints: $1$) The only thing that matters is the component of $(1,1,1)$ in the $(1,0,1)$ direction.

$2$) Consider the function on $[0,1]$ and quote a standard theorem.

$3$) The vector space $V$ has dimension $6$.

André Nicolas
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  • @ Andre, but the answer to the Question 3) is $2$, not $6$. – Fischer Oct 29 '12 at 19:01
  • @Fisher: of course it is. This was a hint. The image of the mapping has dimension $4$, the space $V$ has dimension $6$, so the dimension of the kernel (what is asked for) is $6-4$. – André Nicolas Oct 29 '12 at 20:18
  • Dear @AndréNicolas, how do you know $\ker(T)=\dim(V)-\dim(W)$? Thank you. – Lion Jul 12 '14 at 01:09
  • General and important theorem: If $T$ is any linear mapping from a space $V$ to a space $W$, the dimension of the kernel plus the dimension of the image is equal to the dimension of $V$. – André Nicolas Jul 12 '14 at 01:24