Dr. Mark V. Sapir

Homework due Class 18.

1. Find infinitely many non-proportional functions f(x) in C[0,1] which form angle Pi/3 with the function x².
2. Given the 3-vector A=(1, 2, 3), find two non-zero 3-vectors B and C such that B is orthogonal to A and C, and C is orthogonal to A.

1. Consider the set R² of all 2-vectors. Define a dot product in R² in the following way:
   
   (a,b)(c,d)=ac+bd+(1/2)ad+(1/2)bc
   

   Prove that this dot product satisfies the properties of dot products in Euclidean vector spaces and thus R² with this dot product is a Euclidean vector space.

2. 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 parallelogram.
3. How would you define the sphere of radius r with center A in an arbitrary vector space? Find all constant functions which belong to the sphere of radius 1 with the center x² in C[0,1].

Bonus problems

1. (14 points) Prove or disprove that every linear operator in R² is a composition (product) of a symmetry, a rotation and a dilation.
2. (7 points) Let us define a dot product on R² by the following formula:
   
   (a,b)(c,d)=pac+qbd+rad+sbc
   

   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.

Solutions

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Class 18

Homework due Class 19.

1. 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)
   

2. Is there a solution of the equation T(v)=(1,1,1,1) where T is the operator defined in Problem 1 (use Maple)?
3. Find the standard matrix for the reflection of R² about the axis with direction (1,2).

Solutions

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Class 19

Solutions

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Class 20

1. 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.
   

2. Find the standard matrix of the linear transformation of R³ which reflects every vector V about the plane
   ax+by+cz=0

Solutions

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Test 2. Fall '96

Solutions

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Class 21

1. 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).
2. Find the standard matrix of a linear operator in R² which is the composition of the rotation counterclockwise through angle Pi/3, the reflection about the axes y=5x and the projection on the line y=-2x.

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Class 22

Solutions

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