Question

1. [10] Let ~x ∈ R n with ~x 6= ~0. For each ~y ∈ R n , recall that perp~x(~y) = ~y − proj~x(~y).

(a) Show that perp~x(~y + ~z) = perp~x(~y) + perp~x(~z) for all ~y, ~z ∈ R n .

(b) Show that perp~x(t~y) = tperp~x(~y) for all ~y ∈ R n and t ∈ R.

(c) Show that perp~x(perp~x(~y)) = perp~x(~y) for all ~y ∈ R n

Answer #1

Prove: Let x,y be in R such that x < y.
There exists a z in R such that x < z <
y.
Given:
Axiom 8.1. For all x,y,z in
R:
(i) x + y = y + x
(ii) (x + y) + z = x + (y + z)
(iii) x*(y + z) = x*y + x*z
(iv) x*y = y*x
(v) (x*y)*z = x*(y*z)
Axiom 8.2. There exists a real number 0 such that
for all...

Let X ∼ N(0, 1), ∼ N(0, 1). Let Y = 1.75 + 2X + . Generate 100
samples for (X, Y ). Use the generated data to fit a linear
regression. (a) Report the fitted coefficients and intercept. (b)
Draw a scatter plot of (X, Y ). Add the fitted line and the real
line to the scatter plot with different color.
Using R

1. Let W be the set of all [x y z}^t in R^3 such that xyz = 0.
Is W a subspace of R^3?
2. Let C^0 (R) denote the space of all continuous real-valued
functions f(x) of x in R. Let W be the set of all continuous
functions f(x) such that f(1) = 0. Is W a subspace of C^0(R)?

Let x = [1, 1]T , y = [1, 1]T ∈ R 2 and let f : R 2 =⇒ R 2 with
f(z) =z1.x + z2.y for any z = [z1, z2] T ∈ R 2 . Further, z = g(r)
= [r 2 , r3 ] where r ∈ R . Show how chain rule is applied here
giving major steps of the calculation, write down the expression
for ∂f ∂r , and also evaluate ∂f/ ∂r at...

1. Let R be the rectangle in the xy-plane bounded by the lines x
= 1, x = 4, y = −1, and y = 2. Evaluate Z Z R sin(πx + πy) dA.
2. Let T be the triangle with vertices (0, 0), (0, 2), and (1,
0). Evaluate the integral Z Z T xy^2 dA
ZZ means double integral. All x's are variables. Thank you!.

B.) Let R be the region between the curves y = x^3 , y = 0, x =
1, x = 2. Use the method of cylindrical shells to compute the
volume of the solid obtained by rotating R about the y-axis.
C.) The curve x(t) = sin (π t) y(t) = t^2 − t has two tangent
lines at the point (0, 0). List both of them. Give your answer in
the form y = mx + b ?...

a. Let →u = (x, y, z) ∈ R^3 and define T : R^3 → R^3 as
T( →u ) = T(x, y, z) = (x + y, 2z − y, x − z)
Find the standard matrix for T and decide whether the map T is
invertible.
If yes then find the inverse transformation, if no, then explain
why.
b. Let (x, y, z) ∈ R^3 be given T : R^3 → R^2 by T(x, y, z) = (x...

1.) Let f ( x , y , z ) = x ^3 + y + z + sin ( x + z ) + e^( x
− y). Determine the line integral of f ( x , y , z ) with respect
to arc length over the line segment from (1, 0, 1) to (2, -1,
0)
2.) Letf ( x , y , z ) = x ^3 * y ^2 + y ^3 * z^...

A function f”R n × R m → R is bilinear if for all x, y ∈ R n and
all w, z ∈ R m, and all a ∈ R: • f(x + ay, z) = f(x, z) + af(y, z)
• f(x, w + az) = f(x, w) + af(x, z) (a) Prove that if f is
bilinear, then (0.1) lim (h,k)→(0,0) |f(h, k)| |(h, k)| = 0. (b)
Prove that Df(a, b) · (h, k) = f(a,...

Find all lambda such that the system of equations
-6⋅x+(-10)⋅y+6⋅z= lambda ⋅x, 5⋅x+9⋅y+1⋅z= lambda ⋅y, 0⋅x+0⋅y+5⋅z=
lambda ⋅z has a non-zero solution (x,y,z).

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 minutes ago

asked 5 minutes ago

asked 6 minutes ago

asked 8 minutes ago

asked 13 minutes ago

asked 29 minutes ago

asked 33 minutes ago

asked 37 minutes ago

asked 40 minutes ago

asked 43 minutes ago

asked 51 minutes ago

asked 58 minutes ago