Dr. Mark V. Sapir
Homework due Class 18.
"Proof" homework due Class 20
- Find infinitely many non-proportional functions f(x) in C[0,1]
which form angle Pi/3 with the function x2.
- Given the 3-vector A=(1, 2, 3), find two non-zero
3-vectors B and C such that B is
to A and C, and C is orthogonal to A.
- Consider the set R2 of all 2-vectors. Define
a dot product in R2 in the following way:
Prove that this dot product
of dot products in Euclidean vector spaces and thus
R2 with this dot product is a Euclidean vector space.
- Let V be a Euclidean vector space. A parallelogram is a quadruple
of vectors of the form (A,B,A,B). Define the diagonals of a parallelogram
and prove that the sum of squares of the lengths of
diagonals is equal to the sum of squares of the lengths of the sides of the
- How would you define the sphere of radius r with center A in
vector space? Find all constant functions which belong to the
sphere of radius 1 with the center x2 in C[0,1].
- (14 points) Prove or disprove that every linear operator in R2
is a composition (product) of a symmetry, a rotation and a dilation.
- (7 points) Let us define a dot product on R2 by the following formula:
where p,q,r,s are numbers. Prove that R2 with this dot product is a
Euclidean space if and only if r=s, p> 0, q> 0, and r2-pq< 0.
Homework due Class 19.
- Find the standard matrix of a linear operator R4-->
R4 defined by the formula
T(a,b,c,d)=(4a-3b+4c-d, 3a-5b+d, a+d, b+d)
- Is there a solution of the equation T(v)=(1,1,1,1) where T is the operator
defined in Problem 1 (use Maple)?
- Find the standard matrix for the reflection of R2 about the axis with
Proof homework due Class 23
- Suppose that f is the function from R3 to R3 which rotates every vector
A counterclockwise about some axis L.
a) Show that f is a linear transformation.
b) Show that the column-vectors in the standard matrix A of f are pairwise
orthogonal and their lengths are equal to 1.
- Find the standard matrix of the linear transformation of R3 which
reflects every vector V about the plane
Test 2. Fall '96
Computational homework due Class 22
- Find the standard matrix of a linear transformation from R3 to R2 which
takes (1,2,3) to (1,2), (2,0,1) to (2,0) and (1,2,4) to (1,1).
- Find the standard matrix of a linear operator in R2 which is the
composition of the rotation counterclockwise through angle Pi/3, the
about the axes y=5x and the projection on the line y=-2x.