Math Is Fun Forum

Let S = {(1234),(4321)}
and W the subspace of R4 generated by S.

Find an orthogonal basis of the orthogonal complement W^ of W
Find the shortest distance from u+(12-21) to W^

Obviously the vectors of S are meant to be written as one colum instead of 4.
Thanks.

Let B = {u1,u2} be a basis of a vector space V
Show that C = {v1 = 2u1 - 3u2, v2 = 2u1 + 2u2} is also a basis of V.

Also, what is the change of basis matrix from B to C?

Thanks very much for any help you give me.

Let R1 denote the rectangle [0, 5] × [–4, 4], R2 the rectangle [0, 5] × [0, 4], and R3 the
rectangle [–5, 0] × [–4, 0]. Suppose that ∫∫R2 f (x, y)dA = 10 , that ∫∫R3 f (x, y)dA = 24
and that f (−x, y) = − f (x, y) for all (x, y). Evaluate ∫∫R1 f (x, y)dA

really not sure on the approach here.

On a certain mountain, the elevation z above a point (x, y) in a horizontal xy-plane
at sea level is z = 2500 - x^2/10 - y^2/5   metres.  The positive x-axis points east; the
positive y-axis points north. Suppose that a climber is at the point (15, –10, 2457.5)

if the climber moves southease, what rate will the climber begin to descend at?

i calculated the gradients in the zx direction and in the zy direction, but i'm not sure what the continuation is.

Which spring requires more material to make it, one of radius 5 cm and height 4 cm
that makes three complete turns or one of radius 3 cm and height 4 cm that makes
five complete turns?

i thought i did it right but logically my answer can't be.