Dr. Mark V. Sapir

Class 23

Exercises

Homework due Class 24.

Is it true that the set of all continuous functions f(x) on [0,1] with f(1/2)=1 is a subspace of C[0,1]?
Is it true that the set of all n by n matrices A with trace(A)=0 is a subspace of the vector space of all n by n matrices?
Is it true that the map which takes every function f(x) from C[0,1] to the number f(1) is a linear transformation from C[0,1] to R?
Is it true that the map which takes every function f(x) from C[0,1] to the integral from 0 to 1 of f(x) is a linear transformation from C[0,1] to R.

"Proof" homework due Class 26

Proof the theorem about linear transformations of arbitrary vector spaces.

Solutions

Class 24

Exercises

Homework due Class 25.

Find the kernel and range of the following linear transformations:

The transformation from C[0,1] to R which takes every function f(x) to the number f(1).
The operator on the space of 2 by 2 matrices to itself which takes every matrix

[ a b ]

[ c d ]

to the product
```
  
                          [1 2]   [a b]
                          [   ] * [   ]
                          [0 1]   [c d]
```
(Hint: use the fact that the matrix

[ 1 2 ]

[ 0 1 ]

is invertible.).

In these problems, you do not have to prove that the maps are in fact linear transformations.

Class 25

Exercises

Bonus problems. Due Class 33

(5 points). Prove every rotation in R² is a product of two reflections.
(7 points).Suppose that T is a linear operator in the Euclidean vector space V. Assume that T preserves lengths of vectors that is for every vector v the length of T(v) coincides with the length of v. Prove that T preserves dot products that is for every two vectors u and v the dot product <u,v > is equal to the dot product of T(u) and T(v).
(5 points). Prove or disprove that every linear operator in R² which preserves lengths is a product of several reflections about lines.
(7 points). Prove or disprove that every linear operator in R³ which preserves lengths is a product of several reflections about planes in R³.

Solutions

Class 26

Exercises

Homework due Class 27.

Check if the vector (1,2,3,4) belongs to the column space of the matrix

[ 1 2 3 ]

[ 4 5 6 ]

[ 7 8 9 ]

[ 0 1 2 ]
Find the null space of the matrix

[ 1 2 3 4 5 ]

[ 2 3 4 5 6 ]

[ 1 0 2 0 6 ]
Let p₁=x³+x+1, p₂=2x³-x², p₃=x²+x+1, q₁=x³-1, q₂=x³-x²+1, q₃=x³+x be polynomials. Is it true that span{p₁,p₂,p₃}=span{q₁,q₂,q₃}?
Find the kernel of the linear transformation from R⁴ to R² with the following standard matrix:

[ 1 2 3 4 ]

[ 2 1 3 4 ]
Is it true that the range of the transformation in the previous problem coincides with R²?

Solutions

Class 27

Exercises

Homework due Class 28.

Check whether the following vectors in R⁵ are linearly independent. If they are not linearly independent, find one of them which is a linear combinationof the others.
1. (1,2, 3, 4, 5), (2,3,4,5,6), (1,0,0,0,1), (2, 5, 7, 9, 10);
2. (2,1,1,1,1), (2,1,4,4,4), (3,3,3,1,1).
Are these functions linearly independent as elements of C[0,1]: f(x)=x, g(x)=e^x, h(x) = e^2x.

Solutions

Class 28

Solutions

Test 2

Solutions

Class 29

Exercises

Homework due Class 30.

For which real values of t the following vectors form a linearly independent set in R⁴:
(t, 2t, 3 4), (1, 2, 3t, 4t), (t, 2, 3t, 4), (t, 2, 3, 4t).
Show that the following set of functions from C[0,1] is linearly independent: x, xe^x, x²e^x, x³e^x.

"Proof" homework due Class 31

Prove that

If a subset S of a vector space V contains 0 then S is linearly dependent.
A set S with exactly two vectors is linearly dependent if and only if one of these vectors is a scalar multiple of the other.
If a subset of a set S is linearly dependent then the whole set is linearly dependent

Class 30

Solutions

Class 31

Exercises

Computational homework due Class 32

For which values of t the following 4-vectors form a basis of R⁴: (1,t,3,4), (t, t, 3, 4t), (1,t, 3t, 4), (1, 1, 3, 4t) ?
Find a basis in R⁴ containing the following vectors: (1,1,2,2), (2,3,3,3).
Find a basis of R³ contained in the following set of vectors: (1, 1, 1), (2, 2, 2), (3, 3, 3), (1, 2, 3), (2, 3, 4), (3, 2, 1).
What is the dimension of the subspace of R⁴ spanned by the vectors (1,2,3,4), (1,1,1,1), (3, 4, 5, 6), (5, 7, 9, 11)?

"Proof" homework due Class 34

Prove that if vectors v₁,...,v_n span a vector space V then the vectors v₁-v₂, v₂-v₃, ..., v_n-1-v_n, v_n span V.
Prove that if V is a subspace of a vector space W then dim(V) does not exceed dim(W).
Prove that if V is a subspace of W, dim(V)=dim(W) and W is finite dimensional then V=W.
Let P_n be the space of polynomials of degree at most n. Prove that for every number a the set {1, (x-a), (x-a)²,...,(x-a)ⁿ} is a basis of P_n and the coordinates of every polynomial f(x) from P_n in this basis are (f(a), f'(a), 1/2 f''(a), 1/6 f'''(a),..., (1/(n!)) f⁽ⁿ⁾ (a) ).

Bonus Problems

(5 points) Let a₁, a₂,..., a_k be a linearly independent system of vectors in a vector space V. Find all bases contained in the following system of vectors: b₁=a₁-a₂, b₂=a₂-a₃,...,b_k-1=a_k-1-a_k, b_k=a_k-a₁.
(5 points) In which case a system of vectors spanning a vector space V has a unique subsystem which is a basis of V?
(7 points) Let A be a n by n matrix. Suppose that for every i=1,...,n the absolute value of the diagonal entry A[i,i] is bigger than the sum of absolute values of other entries in the same row. Prove that the column vectorsof this matrix are linearly independent.

Solutions

Class 32

Exercises

Computational homework due Class 33

Find the rank of the set of vectors in R⁵ usng only the definition of the rank: (1,2,3,4,5), (6,7,8,9,10), (1,1,1,1,1), (2,2,2,2,2), (5,4,3,2,1), (6,5,4,3,2).
Find the row rank and the column rank of the following matrix using only the definition of the rank:

[ 2 3 4 5 6 ]

[ 1 1 1 1 1 ]

[ 0 0 0 2 3 ]
Find the rank of the following set of polynomials: x³+x²+x, x³+x+1, x³+x²+x, 2x³+x²+x+1.

Solutions

Class 33

Solutions

Class 34

Solutions

Class 35

Exercises

Computational homework due Class 36

Find the basis of the orthogonal complement of the subspace of R⁴ spanned by the vectors (1,2,3,4) and (2,3,1,1).
For which values of t the following matrices form a basis in M_2,2:

[ t 2t ]

[ 2 3t ]

[ 1 2 ]

[ 2t 3 ]

[ 1 2t ]

[ t+1 t+2 ]

[ 1 t+1 ]

[ 2 2t+1 ]

"Proof" homework due Class 37

Let {v₁,...,v_n} be a basis of a vector space V. Let s₁,...,s_m be vectors of V and we have that

s₁=a₁₁ v₁ +...+ a_1n v_n
s₂=a₂₁ v₁ +...+ a_2n v_n
............................
s_m=a_m1 v₁ +...+ a_mn v_n

for some numbers a_ij. Consider the following n-vectors:

t₁ = (a₁₁, ..., a_1n)
t₂ = (a₂₁, ..., a_2n)
..........................
t_m = (a_m1, ..., a_mn)

Prove that the set {s₁,...,s_m} is linearly independent if and only if the set {t₁,...,t_m} is linearly independent.

Solutions

Class 36

Solutions

Class 37

Exercises

Homework due Class 38.

Find an orthogonal basis of the subspace of C[0,1] spanned by the functions sin(x), cos(x), x.
Find the distance from the function e^x to the subspace spanned by the functions sin(x), cos(x), x in C[0,1].

Solutions

Class 38

Exercises

Homework due Class 39.

Let A be the following matrix:

[ 1 2 3 ]

[ 3 4 5 ]

[ 2 1 3 ]

[ 3 3 6 ]

[ 4 6 8 ]

Find a vector v=(x,y,z) which makes Av closest to the vector (1,2,3,4,5).

Solutions

Class 39

Solutions

Test 3

Solutions

Class 40

Exercises

Problems about eigenvectors and eigenvalues

"Proof" problems.
1. Let v and w be eigenvectors of a linear operator T which correspond to different eigenvalues a and b. Prove that v and w are not proportional.
2. Let v₁, ..., v_k be eigenvectors of T corresponding to pairwise different eigenvalues a₁,...,a_k. Prove that v₁,...,v_k are linearly independent (if you cannot solve this problem, see the book, chapter 7).
3. Prove that the characteristic polynomials of two similar matrices are equal.
4. Let A be a square matrix such that Aⁿ=0 for some n. Prove that all eigenvalues of A are equal to 0 (use the definition of an eigenvalue).
5. Prove that A and A^T have the same eigenvalues (here A is a square matrix).
6. Prove that if a matrix A is similar to a diagonal matrix then A² is also similar to a diagonal matrix.
7. Prove that if two matrix are similar and one of them is invertible then the other one is invertible too.
8. Prove that similar matrices have the same determinants.
9. Prove that similar matrices have the same traces.
10. (Bonus, 10 points). Prove that if A is a symmetric matrix then eigenvectors corresponding to different eigenvalues are orthogonal.
11. (Bonus, 20 points). Prove that if the standard matrix A of a linear operator T in Rⁿ is symmetri then T has a basis of eigenvectors.
12. (Bonus, 7 points). Prove that an upper triangular matrix A is similar to a diagonal matrix if and only if A is diagonal.
Computational problems
1. Let V be the space of all polynomials of degree at most 3. Let T be the linear operator in V which takes each polynomial to its derivative. Find eigenvalues and eigenvectors of this operator.
2. Let T be the linear operator in R² with the following standard matrix:
  
  [ 2 2 ]
  
  [ 8 2 ]
  
  Find the basis where this operator has a diagonal matrix.
3. Find all eigenvectors and eigenvalues of the following matrix:
  
  [ 4 0 1 ]
  
  [ 2 3 2 ]
  
  [ 1 0 4 ]
  
  solution.
4. Determine whether the linear operator in Rⁿ with the following standard matrix:
  
  [ 3 0 2 ]
  
  [ 0 2 0 ]
  
  [ 0 1 2 ]
  
  has a basis of eigenvectors. Hint: find the eigenvalues of this matrix; look how many lineraly independent eigenvectors correspond to each eigenvalue (find the general solution of the corresponding system of equations); find out if the total number of the linearly independent vectors corresponding to all the eigenvalues is equal to the dimension of R³.
5. Determine whether the following matrix is similar to a diagonal matrix.
  
  [ 1 0 0 ]
  
  [ 1 1 0 ]
  
  [ 1 1 1 ]

Solutions

The Final Test. Spring 96.

To see the test, click here.

The Final Test. Fall 96.

To see the test, click here. To see the solutions, click here.