Linear Algebra Questions - Another approach to the proof of the Cauchy-Schwarz Inequality is

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Question 1

Another approach to the proof of the Cauchy-Schwarz Inequality is suggested by the figure below, which shows that in R2 or R3, ||proju(v) || <= ||v||. Show that this is equivalent to the Cauchy- Schwarz Inequality. * The R2 and R3 represent all real numbers in 2 and 3 dimension if that is unclear.

___________

proju(v) u

I know the drawing isn't great but just imagine a right triangle. I took a picture and put it as a file if that will be easier. You will need properties of the absolute value function for this. Question 2 a. Describe all possible values of and that will make the following addition table true. For example, the table tells us that we would need . a+c=2 b. It is a geometric fact that there is a unique parabola through any three noncollinear points (points that don’t all lie on the same line. For the set of points below, find a parabola with an equation of the form ax^2 + bx + c that passes through the given points. Sketch the resulting parabola to check the validity of your answer. (0,1 ) (-1,4 ) (2,1 )

Question 3 Let u= and v = picture these two matricies as u sub 1 like u1

For some reason the sub button won’t allow me to use it inside a matrix. Show that the area A of the parallelogram determined by u and v is given by A = |det |

Hint write u and v as and

There is three questions. Thanks for your help, if you need clarification just ask.