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Linear Algebra for Team-Based Inquiry Learning: PREVIEW Edition — Instructor Version

Steven Clontz, Drew Lewis

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Section 2.1 Linear Combinations (EV1)

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Subsection 2.1.1 Warm Up

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Activity 2.1.1.

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Discuss which of the vectors u→=[1−12] and v→=[03−1] is a solution to the given vector equation:
x1[−123]+x2[2−10]+x3[1−11]=[−115]
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Subsection 2.1.2 Class Activities

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Note 2.1.2.

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We’ve been working with Euclidean vector spaces of the form
Rn={[x1x2⋮xn]|x1,x2,…,xn∈R}.
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There are other kinds of vector spaces as well (e.g. polynomials, matrices), which we will investigate in Section 3.5. But understanding the structure of Euclidean vectors on their own will be beneficial, even when we turn our attention to other kinds of vectors.
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Likewise, when we multiply a vector by a real number, as in −3[1−12]=[−33−6], we refer to this real number as a scalar.
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Definition 2.1.3.

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A linear combination of a set of vectors {v→1,v→2,…,v→n} is given by c1v→1+c2v→2+⋯+cnv→n for any choice of scalar multiples c1,c2,…,cn.
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For example, we can say [305] is a linear combination of the vectors [1−12] and [121] since
[305]=2[1−12]+1[121].
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Definition 2.1.4.

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The span of a set of vectors is the collection of all linear combinations of that set:
span{v→1,v→2,…,v→n}={c1v→1+c2v→2+⋯+cnv→n|ci∈R}.
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For example:
span{[1−12],[121]}={a[1−12]+b[121]|a,b∈R}.
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Activity 2.1.5.

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Consider span{[12]}.
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(a)
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Sketch the four Euclidean vectors
1[12]=[12],3[12]=[36],0[12]=[00],−2[12]=[−2−4]
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in the xy plane by placing a dot at the (x,y) coordinate associated with each vector.
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(b)
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Sketch a representation of all the vectors belonging to
span{[12]}={a[12]|a∈R}
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in the xy plane by plotting their (x,y) coordinates as dots. What best describes this sketch?
  1. A line
  2. A plane
  3. A parabola
  4. A circle
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Remark 2.1.6.

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It is important to remember that
{v→1,v→2,…,v→n}≠span{v→1,v→2,…,v→n}.
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For example,
{[1−12],[121]}
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is a set containing exactly two vectors, while
span{[1−12],[121]}={a[1−12]+b[121]|a,b∈R}
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is a set containing infinitely-many vectors.
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Activity 2.1.7.

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Consider span{[12],[−11]}.
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(a)
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Sketch the following five Euclidean vectors in the xy plane.
1[12]+0[−11]=?0[12]+1[−11]=?1[12]+1[−11]=?
−2[12]+1[−11]=?−1[12]+−2[−11]=?
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(b)
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Sketch a representation of all the vectors belonging to
span{[12],[−11]}={a[12]+b[−11]|a,b∈R}
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in the xy plane. What best describes this sketch?
  1. A line
  2. A plane
  3. A parabola
  4. A circle
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Activity 2.1.8.

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Sketch a representation of all the vectors belonging to span{[6−4],[−32]} in the xy plane. What best describes this sketch?
  1. A line
  2. A plane
  3. A parabola
  4. A cube
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Activity 2.1.9.

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Consider the following questions to discover whether a Euclidean vector belongs to a span.
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(a)
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The Euclidean vector [−1−61] belongs to span{[10−3],[−1−32]} exactly when there exists a solution to which of these vector equations?
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  1. x1[−1−61]+x2[10−3]=[−1−32]
  2. x1[10−3]+x2[−1−32]=[−1−61]
  3. x1[−1−32]+x2[−1−61]+x3[10−3]=0
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(b)
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Use technology to find RREF of the corresponding augmented matrix, and then use that matrix to find the solution set of the vector equation.
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(c)
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Given this solution set, does [−1−61] belong to span{[10−3],[−1−32]}?
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Observation 2.1.10.

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The following are all equivalent statements:
  • The vector b→ belongs to span{v→1,…,v→n}.
  • The vector b→ is a linear combination of the vectors v→1,…,v→n.
  • The vector equation x1v→1+⋯+xnv→n=b→ is consistent.
  • The linear system corresponding to [v→1…v→n|b→] is consistent.
  • RREF[v→1…v→n|b→] doesn’t have a row [0⋯0|1] representing the contradiction 0=1.
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Activity 2.1.11.

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Consider the following claim:
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[−62−6]is a linear combination of the vectors [102],[306],[204], and [−41−5].
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(a)
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Write a statement involving the solutions of a vector equation that’s equivalent to this claim.
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(c)
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Since your statement was true, use the solution set to describe a linear combination of [102],[306],[204], and [−41−5] that equals [−62−6].
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Activity 2.1.12.

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Consider the following claim:
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[−5−1−7] belongs to span{[102],[306],[204],[−41−5]}.
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(a)
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Write a statement involving the solutions of a vector equation that’s equivalent to this claim.
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(b)
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Explain why the statement you wrote is false, to conclude that the vector does not belong to the span.
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Subsection 2.1.3 Individual Practice

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Activity 2.1.13.

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Before next class, find some time to do the following:
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(a)
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Without referring to your activity book, write down the definition of a linear combination of vectors.
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(b)
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Let u→=[120] and v→=[−130]. Write down an example w1→=[???] of a linear combination of u→,v→. Then write down an example w2→=[???] that is not a linear combination of u→,v→.
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(c)
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Draw a rough sketch of the vectors u→=[120], v→=[−130], w1→=[???], and w2→=[???] in R3.
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Subsection 2.1.4 Videos

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Figure 5. Video: Linear combinations
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Exercises 2.1.5 Exercises

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Subsection 2.1.6 Mathematical Writing Explorations

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Exploration 2.1.14.

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Suppose S={v1→,…,vn→} is a set of vectors. Show that v0→ is a linear combination of members of S, if an only if there are a set of scalars {c0,c1,…,cn} such that z→=c0v0→+⋯+cnvn→. We can do this in a few parts. I’ve used bullets here to indicate all that needs to be done. This is an "if and only if" proof, so it needs two parts.
  • First, assume that 0→=c0v0→+⋯+cnvn→ has a solution, with c0≠0. Show that v0→ is a linear combination of elements of S.
  • Next, assume that v0→ is a linear combination of elements of S. Can you find the appropriate {c0,c1,…,cn} to make the equation z→=c0v0→+⋯+cnvn→ true?
  • In either of your proofs above, does the case when v0→=z→ change your thinking? Explain why or why not.
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Subsection 2.1.7 Sample Problem and Solution

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Sample problem Example B.1.5.
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