Final Practice Exercises Part I
Math 4581 Autumn 2013
1) Let H be the plane in R3 whose equation is: x 5y + 9z = 0.
a) Verify that H is a vector subspace of R3 .
b) Give a basis B for H .
c) Extend the basis B to a basis B for R3 .
2) Consider H = cfw_(x,
Final Practice Exercises Part II
Math 4581 Autumn 2013
1) Espress f (r, s, t) = r2 s + r2 t + rs2 + s2 t + rt2 + st2 as polynomial in
the symmetric elementary polynomials s1 , s2 and s3 .
2) Show that the polynomial f (r, t) = r2 + t2 2r2 t 2rt2 is symmet
Final Practice
Math 4581 Autumn 2013
1) Let H be the plane in R3 whose equation is x 6y = 0.
a) Verify that H is a vector subspace of R3 .
b) Give a basis B for H .
c) Extend B to a basis B for R3 .
d) Let L be the following vector subspace of R3
L := cfw
Midterm 1 practice test
Math 4581 Autumn 2013
1) Let H be the plane in R3 whose equation is x y = 0.
a) Verify that H is a vector subspace of R3 .
b) Give a basis B for H .
c) Extend B to a basis B for R3 .
2) Let H and K be the vector subspaces of R3 den
Midterm 2 practice test
Math 4581 Autumn 2013
1) Let P be a regular hexagon in R2 centered at the origin (0, 0). Prove:
if f Isom(P ) and f maps each long diagonal of P to itself (but not necessarily pointwise), then f = I or f = I .
2) Let T be a tetrahe