## Minimizing essup of difference of functions in $C^0$ and $L^\infty$ November 30

Let $g\in C^0[0,1]$. Minimize $||f-g||_{\infty}$ for all $f\in L^\infty [0,1]$ such that $\int_0^1 fdx = 0$. Considering $|\int_0^1 g-f dx| \ge |\int_0^1 gdx|$ by the requirement on $f$. In the trivial case $\int_0^1 g dx = 0$ so we can pick $f=g$. N

## Show that linear transformation is not surjective November 30

Given the matrix, $$M = \begin{bmatrix}1&7&9&3\\2&15&19&8\\7&52&66&27\\3&4&10&-24\end{bmatrix}$$Show that the linear transformation $T_m: \mathbb R^4 \to \mathbb R^4$ defined by the multiplication of column

## Finding the distance between a plane and $(0,0,0)$ November 30    1

Given the lines:$\frac{x+1}{4} = \frac{y-3}{1} = \frac{z}{k}$ and $\frac{x-1}{3} = \frac{y+2}{-2} = \frac{z}{1}$ that lie on the same plane.How can I find the parameter $k$ ? (I guess I'll be able to calculate the distance between the resulting pl

## Distance between planes November 30    4

Find the distance between the planes $$x + 2y +2z = 4$$ $$z= -\frac12 (x-1)-(y-2)+3$$First of all how do you check if they are parallel? The integers in plane two are leading me astray? How do I handle those integers. I know that using the dot produc

## How to determine the curve November 30    4

In the figure above, segment $PQ$ is determined by two points: $P: (t,0)$ and $Q: (1,t)$, where $t\in [0,1]$ continuously increases and decreases between $0$ and $1$.Then this gives a close ...

## Let L be the line of intersection of the planes $cx + y + z = c$ and $xcy + cz = -1$, where c is a real number. November 30    1

Find symmetric equations for $L$As the number $c$ varies, the line $L$ sweeps out a surface $S$. Find an equation for the curve of intersection of $S$ with the horizontal plane $z = t$ (the trace of S in the plane $z = t$) I got the symmetric equatio

## Partial derivative not parallel to $x$ or $y$ axis November 30    3

In attached example I want to find slope along $x=y$ direction. I know how to find partial derivatives when we are parallel to $x$ axis (keep $y$ constant and take partial derivative with respect to $x$) or $y$ axis, but I don't know how to get find

## Prove that $\sum\limits_{i=1}^n 2i\binom{2n}{n-i}= n\binom{2n}{n}$ November 30    2

Let n be a positive integer.Prove that $$\sum_{i=1}^n 2i\binom{2n}{n-i}= n\binom{2n}{n}$$ Here's an alternative that requires a little less of a leap to get started, but a little more algebra later. Instead of pulling $2i=(n+i)-(n-i)$ out of thin air

## Partial derivative of a two degree plane curve November 30    1

Partial derivative of a two degree plane curve, with respect to x and y when taken, we get two linear expressions in x and y. If we equate it to zero, and solve them simultaneously ...we get a point. Why this point is the point of intersection of the

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