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Back to The span of a set of vectors

Exercise 1

Avg difficulty 2.5

In this exercise, we will consider the span of some sets of two- and three-dimensional vectors.

  1. Consider the vectors v1=[1−2],v2=[43]\vvec_1 = \twovec{1}{-2}, \vvec_2 = \twovec{4}{3}.

    1. Is b=[21]\bvec = \twovec{2}{1} in Span{v1,v2}\laspan{\vvec_1,\vvec_2}?

    2. Give a geometric description of Span{v1,v2}\laspan{\vvec_1,\vvec_2}.

  2. Consider the vectors v1=[213],v2=[−202],v3=[61−1]\vvec_1=\threevec{2}{1}{3}, \vvec_2=\threevec{-2}{0}{2}, \vvec_3=\threevec{6}{1}{-1}.

    1. Is the vector b=[−10−15]\bvec=\threevec{-10}{-1}{5} in Span{v1,v2,v3}\laspan{\vvec_1,\vvec_2,\vvec_3}?

    2. Is the vector v3\vvec_3 in Span{v1,v2,v3}\laspan{\vvec_1,\vvec_2,\vvec_3}?

    3. Is the vector b=[33−1]\bvec=\threevec{3}{3}{-1} in Span{v1,v2,v3}\laspan{\vvec_1,\vvec_2,\vvec_3}?

    4. Give a geometric description of Span{v1,v2,v3}\laspan{\vvec_1,\vvec_2,\vvec_3}.

Austin, Understanding Linear Algebra, §2.3 The span of a set of vectors, Exercise 1

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Adapted from Understanding Linear Algebra © 2023 David Austin. Licensed under CC BY 4.0. Source: Understanding Linear Algebra. Imported at revision 3e6dced.