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    EE472 - Control Theory II - Summer 2018 Homework 1: (Only Problems With Will Be Graded)

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    Hussain Salman
    This document contains 19 questions regarding linear algebra concepts such as linear independence, basis, row echelon form, rank, null space, column space, and solving systems of linear equations. The questions involve determining if sets of vectors are linearly independent, finding bases, reducing matrices to row echelon form, computing matrix ranks, solving homogeneous and non-homogeneous systems of equations, and determining spans.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    EE472 - Control Theory II - Summer 2018 Homework 1: (Only Problems With Will Be Graded)

    Uploaded by

    Hussain Salman
    18 views4 pages
    This document contains 19 questions regarding linear algebra concepts such as linear independence, basis, row echelon form, rank, null space, column space, and solving systems of linear equations. The questions involve determining if sets of vectors are linearly independent, finding bases, reducing matrices to row echelon form, computing matrix ranks, solving homogeneous and non-homogeneous systems of equations, and determining spans.

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    This document contains 19 questions regarding linear algebra concepts such as linear independence, basis, row echelon form, rank, null space, column space, and solving systems of linear equations. The questions involve determining if sets of vectors are linearly independent, finding bases, reducing matrices to row echelon form, computing matrix ranks, solving homogeneous and non-homogeneous systems of equations, and determining spans.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    18 views4 pages

    EE472 - Control Theory II - Summer 2018 Homework 1: (Only Problems With Will Be Graded)

    Uploaded by

    Hussain Salman
    This document contains 19 questions regarding linear algebra concepts such as linear independence, basis, row echelon form, rank, null space, column space, and solving systems of linear equations. The questions involve determining if sets of vectors are linearly independent, finding bases, reducing matrices to row echelon form, computing matrix ranks, solving homogeneous and non-homogeneous systems of equations, and determining spans.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 4

    EE472 – Control Theory II – Summer 2018

    Homework 1
    (Only problems with * will be graded)

    Question 1*: Check whether the following sets of vectors are linearly independent or
    not. If they are linearly dependent, then find a non-trivial linear combination of them
    which is zero.
    1 0 1
    3 1 5
    1 1 17 0 1 3
    a) , , , b) 6 , 0 , 12 , c) , , .
    1 3 20 1 1 4
    4 1 9
    2 3 10
    Question 2:
    a) Find a basis for the subspace S of R3 defined by the equation 2x -4y + 5z = 0
    b) Verify that v1 = [4 2 0]T S and find a basis for S which includes v1.

    Question 3: Given the following matrices:

    1 -1 -3 2 -6
    -1 2 -7 4 1
    2 3 -1 1 6
    A1 , A2 3 1 0 0 2
    3 7 1 0 18
    6 1 3 -2 3
    4 6 -2 1 13
    a) Reduce these matrices to their row echelon form.
    b) Reduce these matrices to their reduced row echelon form. Check your answer
    using MATLAB.
    c) Compute the rank of these matrices.

    Question 4: Find i) a basis of the column-space ii) a basis of the row-space and iii) a
    basis of the null-space of the matrices given below and then deduce their ranks.
    1 1 2 0 1 3
    3 2 4 -3 9
    -1 1 -5 9 1 2 1 4 0 3 2
    a) A , b) A -2 1 -5 -5 -20 c) A
    2 -2 10 -13 -2 , 3
    1 2
    4 -1 9 7 34 0 0 0 1 3 0
    7 4 2 -50 4
    5 7 3 -16 -7 3 0 2 4 3 2
    Question 5*:
    a) Find the rank of the following matrix for the different values of a, b and c.
    1 0 0
    c a 4 5 b
    A
    0 b 5 a 4
    0 0 3c
    b) If b [b1 b2 b3 b4 ]T , characterize all vectors b such that Ax=b is solvable.
    c) If a=5 and b=2 and c=0, solve the equation Ax= [4 6 15 0]T
    Question 6:
    a) A vector space V consists of the vectors {v1 , v2 , v3 , v4 } and all their linear
    combinations. Determine the dimension and a basis for V if
    v1 [3 -2 4 5 6]T , v2 [1 6 7 3 2]T v3 [ 17 42 68 39 34]T and
    v4 [6 5 3 2 1]T .
    b) Is the basis you found unique? Justify your answer.
    c) Is it possible to express v [133 393 650 367 321]T as a linear combination of
    {v1 , v2 , v3 , v4 } ? Justify your answer.

    Question 7*: Let the matrix A be such:


    1 -2 13 8 38
    A 2 3 5 -5 -15
    4 7 7 -13 -43
    T
    a) Give a full solution to the equation Ax c 35 -21 -55 .
    b) Find a basis of the column space of A and a basis of the null-space of A.
    c) Find the rank of A.
    d) Find the dimensions of the column space of A and the null-space of A.
    e) Find a basis of the null-space of A.
    f) If b [b1 b2 b3 ]T , characterize all vectors b such that Ax=b is solvable.
    g) If b [63 -35 -93]T , is the equation Ax=b solvable? If yes, find all solutions.
    h) If b [27 26 37]T , is the equation Ax=b solvable? If yes, find all solutions.

    Question 8*: Solve the following sets of linear equations:


    3 x1 x3 12
    3 x1 5 x2 10 x2 2 x3 4 x4 5
    x1 5 x2 2 x3 7
    a) x2 4 x3 5 b) 3 x1 x2 4 x4 12 c)
    13 x1 20 x2 5 x3 8
    3 x1 7 x3 35 3 x1 3 x2 4 x3 4 x4 22
    47 x1 70 x2 17 x3 34

    Question 9: Solve the following set of linear equations:


    x k1 y 8
    3 x 4k 2 y k3
    for the different values of the constants k1 , k 2 and k3 (you need to determine for which
    values of the constants k1 , k 2 and k3 does the set have no solution, many solutions, and
    a unique solution)
    Question 10: Without using the calculator, find the determinant and the inverse (if it
    exists) of each of the following matrices:

    1 0 1 3 2 1
    3 4
    A1 2 1 4 , A2 1 2 1 , A3 .
    2 3
    1 1 9 0 5 6

    Question 11: Determine which of the following subsets are subspaces?


    x x
    a) S 1 /x 2y b) S 2 /x 2y and 2x y
    y y
    x
    x
    c) S 3 /x 2y 1 . d) S 4 y /x 2 y and x 2 y z 0
    y
    z
    Question 12: Check whether the following sets of vectors are linearly independent or
    not. If they are linearly dependent, then find a non-trivial linear combination of them
    which is zero.

    1 1 1 0 1 1 14
    a) , , b) , , c) , , ,
    2 1 2 0 2 2 19
    1 2 1 1 1 1 1
    d) 2 , 4 , e) 2 , 0 , f) 2 , 0 , 4 ,
    3 6 3 1 3 1 9

    Question 13: For which real numbers are the following vectors linearly independent
    T T T
    in R3? u1 1 1 , u2 1 0 , u3 1 1 .
    a b c
    Question 14: Find the conditions on a, b, c such that A
    1 1 1
    a) rank A =1 b) rank A = 2.

    Question 15: Let the matrix A be such:


    1 2 5 1 0
    1 1 4 1 1
    A
    0 1 1 1 0
    1 2 5 0 1

    a) Find a basis of the column space of A, a basis of the row space of A and a basis
    of the nullspace of A.
    b) Find the rank of A.
    c) Find the dimensions of the column space of A and the nullspace of A.
    d) If b [b1 b 2 b3 b 4 ]T , characterize all vectors b such that Ax=b is solvable.
    e) If b [1 1 1 1]T , is the equation Ax=b solvable? If yes, find all solutions.
    f) If b [1 1 1 2]T , is the equation Ax=b solvable? If yes, find all solutions.
    Question 16:
    a) Find the rank of the following matrices:

    0 1 0 4 1 1 1 2 3 4
    A1 0 0 0 A2 3 2 0 A3 0 1 2 2
    0 0 1 1 1 0 0 0 0 1
    b) Find the range spaces and null spaces of the above matrices.

    Question 17:
    a) Find the general solution of the equation:

    1 2 3 4 3
    0 1 2 2 x 2
    0 0 0 1 1
    b) Find the solution that has the smallest Euclidean norm.

    Question 18*:
    a) A vector space V consists of the vectors {v1 , v2 , v3 , v4 } and all their linear
    combinations. Determine the dimension and the basis for V if
    v1 [1 2 3 4 5]T , v2 [1 3 5 6 7]T v3 [1 3 6 8 9]T and
    v4 [1 3 6 10 12]T .
    Is the basis unique? Justify your answer.
    b) Is it possible to express v [5 4 3 2 1]T as a linear combination of
    {v1 , v2 , v3 , v4 } ? Justify your answer.

    Question 19:
    a) Determine whether the vectors v1 [3 1 4]T , v2 [2 3 5]T , v3 [5 2 9]T
    and v4 [1 4 1]T span IR 3 ?
    b) Determine whether the following polynomials span P2 (the set of all polynomials of
    order less than or equal 2)
    p1 1 x 2 x 2 , p2 3 x , p3 5 x 4 x 2 and p4 2 2 x 2 x2

    Question 20:
    a) Determine the dimensions of the following subspaces of R4.
    T
    i. All vectors of the form a b c 0 .
    T
    ii. All vectors of the form a b c d with d = a + b, c = a – b.
    T
    iii. All vectors of the form a b c d with a = b = c = d.
    T
    iv. All vectors of the form a b c d with a = 2b + c, b = d – a,
    c = 2a – b – d.
    b) Find a basis for the space in (a), i, and find the coordinate vector of
    T
    1 1 0 2 with respect to this basis.

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