Linear Algebra Math 310
Spring 2012
Assignment Sheet 2

This sheet specifies a selection of exercises from the text, and in some cases a few additional exercises. The notation 1-33o means the odd problems between 1 and 33, inclusive.  For problems that require you to find the rref of a matrix, feel free to use software such as freemat.  Note: you can find rref at wolframalpha.com by entering an input of the form rref([1,2,3],[4,5,6]).

Section 1.7 1-19o, 21*, 27*, 28*,29,30*, 31 plus these.

  1. Try to complete the following statements of theorems and/or definitions without looking them up in the book:
    1. A set of two vectors is linearly dependent if and only if ...
    2. If a set of vectors in Rn contains the zero vector then ...
    3. For a set of vectors, if the number of vectors in the set is more than the number of entries in each vector, then the set is ...
    4. The columns of a matrix A are linearly independent if and only if the matrix equation Ax = 0 ...
    5. If S = { v1 , ..., vp } is a set of two or more vectors, then S is a linearly dependent set if and only if ...
  2. (*) Prove: If {v1,v2, ..., vn } is a dependent set of vectors, then so is {v1,v2, ..., vn,vn+1 } for any vector vn+1.
  3. (*) Prove: If {v1,v2, ..., vn } is an independent set of vectors, then so is any non-empty subset. To simplify the argument, you may assume that any non-empty subset can be expressed as {v1,v2, ..., vk } for some k with 1 < k < n. Why is that valid?
  4. (*) Make up a definition for linear independence of functions (as opposed to vectors). According to your definition, are the functions f(x) = x and g(x) = x2 linearly independent? What about the functions cos2x, sin2x, and 2-cos2x?

Section 1.8 1, 3, 5, 8, 9,11, 19, 21*, 30*, 31*. Plus the following, all (*)s:

  1. The set F is the set of functions from R to R. Which of the following are linear transformations from F to F? Justify your answers. [Note that these are operations on functions, so you need to be concerned about adding functions (f+g) and multiplying functions by scalars (2f)].
  2. If T is a linear transformation from R2 to R2, and if S is a linear transformation from R2 to R3, it is possible to combine them to form a composite transformation: first apply T and then apply S to the result. That is, map any vector v to S(T(v)). Must this be a linear transformation?
  3. Let T be a linear transformation, and let N be the set of all the vectors which this transformation maps to 0. If u and v are elements of N, show that every vector in the span {u,v} is also in N.

Section 1.9 (Before the exam) 1, 8, 10, 17, plus the following.

Try to complete the following statements of theorems and/or definitions without looking them up in the book:

  1. A linear mapping from R3 to R4 is defined as follows.  T takes the vector [x, y, z] (written as a row rather than a column for convenience), interchanges x and y, doubles z, and then inserts the average of the original x, y, and z as the fourth entry of the output vector.  For example, applying T to the vector [1, 2, 3] results in  [2, 1, 6, 2].  Assuming this is a linear transformation, find its matrix.

Section 1.9 (After the exam) 23*,25-29o, plus the following.

Try to complete the following statements of theorems and/or definitions without looking them up in the book:

  1. A mapping from Rn to Rm is onto if ...
  2. A mapping from Rn to Rm is one-to-one if ...
  3. A linear transformation T from Rn to Rm is one-to-one if and only if the equation T(x) = 0 ...
  4. If T: Rn  Rm is a linear transformation and A is the standard matrix for T, then T is onto if and only if the columns of A ____________ and T is one-to-one if and only if the columns of A ____________

Section 2.1 1-7o,8-13,17-21,(22-24)*,27,28,29*,31,32. Plus these:

  1. Define the following terms: diagonal entries, diagonal matrix, row-column rule, commute, transpose.
  2. Suppose A and B are 2 × 2 matrices. For each of the following equations, write Always if the equation is true for all A and B, Sometimes if the equation is true for some A and B but not for others, and Never if the equation is not true for any A and B. For each equation, give an explanation of your answer.
    1. (A+B)2 = A2 +2AB + B2
    2. (A+B)(A-B) = A2 - B2
    3. (A3)5 = A15
    4. A3A5 = A8
      5.
    5. (AB)3 = A3B3
  3. (*) The blue box on page 102 indicates that the columns of AB must be linear combinations of the columns of A. Show that the reverse is not true. That is, show (by example) that the columns of A need not be linear combinations of the columns of AB.
  4. (*) A matrix A is called symmetric if AT = A. Show that for any matrix A, ATA and AAT are symmetric. Does ATAAT have to be symmetric?
  5. (*) Let A and B be n × n matrices. Suppose that the system Ax = O has a nontrivial solution. Does BAx = O have to have a nontrivial solution? How about ABx = O ?
  6. (*) Prove: If A and B are n × n matrices and if the columns of A are linearly dependent then the columns of BA are linearly dependent as well.

Section 2.2 1-11o, 12, 13, 15,  16*, 21-24, 29, 31, 33*,35. Plus:

  1. Define the terms: invertible, inverse, elementary matrix.
  2. State a theorem that concerns the existence and uniqueness of solutions for a system A x = b under the assumption that A is invertible.
  3. State a theorem that concerns inverses of products and transposes of invertible matrices.
  4. State a theorem that concerns the row operations, row equivalence, and invertibility of a matrix A.
  5. (*) Find the conditions under which a 2 × 2 matrix A satisfies A-1 = AT. Prove that A-1 = AT if and only if your conditions hold.
  6. (*) We say that a square matrix A satisfies a polynomial p(x) = cnxn +cn-1xn-1 + ...+ c1x + c0    if    
    p    (A) = cnAn + cn-1An-1 + ...+ c1A + c0I = 0.
    Suppose A satisfies the polynomial p(x) = x3 + 4x2 - 7x +2. Show that A is invertible, and find a formula for its inverse.

Section 2.3 1-7o,11,12,16-21,25*,26,27*, 33, 35,36.

Determinant Handout: Read the handout and do the problems (from our text) listed at the end.