MA-305 Homework 5

Due at 4:05pm, Thursday, October 23, 1997
Extended to 1:00pm, Friday, October 24, 1997


You may do the calculations necessary for these problems either by hand or with Maple. Please submit a handwritten solution, or email an ASCII/Postscript/html document to the TA. (You may also submit through email which attached your Maple worksheet.)

  1. *Determine whether the given set V is a vector space under the operation +¨ and ר:

    V is the set of all ordered triples of real numbers of the form ( 0, y, z);

    ( 0, y, z) +¨ ( 0, y', z') = ( 0, y + y', z + z')

    and

    c ר ( 0, y, z) = ( 0, 0, cz).

  2. *Determine whether the given set V is a vector space under the operation +¨ and ר:

    V is the set of all polynomials of the form at2 + bt + c , where a, b, and c are real numbers with b = a + 1 ;

    ( a1t2 + b1t + c1 ) +¨ ( a2t2 + b2t + c2 ) = ( a1 + a2 )t2 + ( b1 + b2 )t + ( c1 + c2 )

    and

    r ר ( at2 + bt + c ) = (ra)t2 + (rb)t + c.

  3. *M23 is the vector space of all 2 × 3 matrices under the usual operations of matrix addition and scalar multiplication. Which of the following subset of the vector space M23 are subspaces? The set of all matrices of the form

    (a) [ a b c ] , where b = a + c.
    [ d 0 0 ]

    (b) [ a b c ] , where c > 0.
    [ d 0 0 ]

    (c) [ a b c ] , where a = -2c and f = 2e +d.
    [ d e f ]

  4. *Find a set of vectors spanning the solution space of Ax=0, where

    [ 1 0 1 0 ]
    A =[ 1 2 3 1 ]
    [ 2 1 3 1 ]
    [ 1 1 2 1 ]

  5. *Which of the following sets of vectors in R3 are linearly dependent? For those that are, express one vector as a linear combination of the rest.

    (a) { ( 1, 2, -1 ), ( 3, 2, 5 ) }.
    (b) { ( 4, 2, 1 ), ( 2, 6, -5 ), ( 1, -2, 3 ) }.
    (c) { ( 1, 1, 0 ), ( 0, 2, 3 ), ( 1, 2, 3 ), ( 3, 6, 6 ) }.

  6. *Suppose that S = {v1 , v2 , v3} is a linearly independent set of vectors in a vector space V. Show that T = {w1 , w2 , w3} is also linearly independent, where w1 = v1 + v2 + v3 , w2 = v2 + v3 , and w3 = v3 .

  7. *Which fo the following sets of vectors are bases for R3 ?

    (a) { ( 1, 2, -0 ), ( 0, 1, -1 ) }.
    (b) { ( 1, 1, -1 ), ( 2, 3, 4 ), ( 4, 1, -1 ), ( 0, 1, -1 ) }.
    (c) { ( 3, 2, 2 ), ( -1, 2, 1 ), ( 0, 1, 0 ) }.
    (d) { ( 1, 0, 0 ), ( 0, 2, -1 ), ( 3, 4, 1 ), ( 0, 1, 0 ) }.

  8. Find a basis for the null space of the given matrix A.

    [ 1 2 1 0 ]
    A = [ 2 3 4 7 ]
    [ 3 5 5 7 ]
    [ 3 4 7 14 ]

*: problems from ``Introductory Linear Algebra with Applications'' by Kolman and Hill