Download 2010Fall-LA-AssignmentTheLastOneNo3

Linear Algebra
Homework No. 3
Group A.
𝑣1
𝑣 +𝑣
Problem 1. Let 𝑇 ([𝑣2 ]) = [𝑣1 + 𝑣3 ]. Is T linear operator? Prove or disprove. If it
2
3
𝑣3
is find the transformation(standard) matrix.
1 1 2
0 2 2
Problem 2. Let = 1 , 0 , 1 . Using Gram-Schmidt orthogonalization
1 0 1
[
{ 1] [1] [2]}
method to find an orthonormal basis for span(S).
Problem 3(10pts). Let S be a space of all trigonometric polynomials of the type
𝑦 = 𝐴𝑠𝑖𝑛(𝑥) + 𝐵𝑐𝑜𝑠(𝑥) + 𝐶
𝐴
Each trigonometric polynomial 𝑦 corresponds to the vector [𝐵 ]. This
𝐶
representation of 𝑦 we will denote as [𝑦].
(a) Show that for two trigonometric polynomials y and z
[𝑦 + 𝑧] = [𝑦] + [𝑧]
And for any scalar k
[𝑘𝑦] = 𝑘[𝑦]
(b) Let D be a derivative operator on S. Show D is linear.
(c) Using the canonical basis for 𝑅 3 find the transformation matrix for D.
(d) Using Linear Algebra methods find all trigonometric polynomials which
are solution for the ODE.
𝑦 ′′ − 𝑦 = 1
Problem 4. Using Linear Algebra method find all polynomials of degree equal
or less than 3 which are solution for
𝑦 ′′ + 𝑦 ′ = 4𝑥 2 − 1
Group B.
𝑣1
𝑣4
𝑣2
𝑣1
Problem 1. Let 𝑇 ([𝑣 ]) = [𝑣 ]. Is T linear operator? Prove or disprove. If it is
2
3
𝑣4
𝑣3
find the transformation(standard) matrix.
1 1 1
0 2 0
Problem 2. Let = 1 , 0 , 1 . Using Gram-Schmidt orthogonalization
1 0 2
{[1] [1] [1]}
method to find an orthonormal basis for span(S).
Problem 3. Let 𝑅𝛼 be the transformation matrix representing rotation for an
angle 𝛼. Prove or disprove:
(𝑅𝛼 )−1 = 𝑅−𝛼
Problem 4. Using Linear Algebra method find all polynomials of degree equal
or less than 3 which are solution for
𝑦 ′′ − 2𝑦 ′ + 𝑦 = 4𝑥 2 − 1
Problem 5. Let T be a linear operator. Prove or disprove:
𝑇(0) = 0.
Group C.
1
1
2
2
Problem 1. Let 𝑣 = [ ] , 𝑢 = [ ]. Find the projection of v onto u.
1
1
1
0
1 1 1
1 2 3
Problem 2. Let = 1 , 1 , 1 . Using Gram-Schmidt orthogonalization
1 2 2
[
{ 1] [1] [1]}
method to find an orthonormal basis for span(S).
Problem 3. Find the matrix R which represent the rotation for 45° . Then
compute 𝑅 4 .
Problem 4. Let u be a vector and T an operator defined by T(u)=2u.
Prove or disprove that T is a linear operator.
1
Problem 5. Let T be a linear operator from 𝑅 into 𝑅 ,such that 𝑇(𝑒1 ) = [2],
1
2
1
1
𝑇(𝑒2 ) = [2], 𝑇(𝑒3 ) = [2]. Find 𝑇 ( [0]).
1
2
1
3
3