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Section 2.2 Linear Combinations (VS2)

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Subsection 2.2.1 Class Activities

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Definition 2.2.1.

A linear combination of a set of vectors {v→1,v→2,…,v→m} is given by c1v→1+c2v→2+⋯+cmv→m for any choice of scalar multiples c1,c2,…,cm.

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 2.2.2.

The span of a set of vectors is the collection of all linear combinations of that set:

span{v→1,v→2,…,v→m}={c1v→1+c2v→2+⋯+cmv→m|ci∈R}.

For example:

span{[1−12],[121]}={a[1−12]+b[121]|a,b∈R}.
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Activity 2.2.3.

Consider span{[12]}.

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(a)

Sketch 1[12]=[12], 3[12]=[36], 0[12]=[00], and −2[12]=[−2−4] in the xy plane.

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(b)

Sketch a representation of all the vectors belonging to span{[12]}={a[12]|a∈R} in the xy plane.

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Activity 2.2.4.

Consider span{[12],[−11]}.

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(a)

Sketch the following linear combinations 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)

Sketch a representation of all the vectors belonging to span{[12],[−11]}={a[12]+b[−11]|a,b∈R} in the xy plane.

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Activity 2.2.5.

Sketch a representation of all the vectors belonging to span{[6−4],[−32]} in the xy plane.

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Activity 2.2.6.

The vector [−1−61] belongs to span{[10−3],[−1−32]} exactly when there exists a solution to the vector equation x1[10−3]+x2[−1−32]=[−1−61].

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(a)

Reinterpret this vector equation as a system of linear equations.

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(b)

Find its solution set, using technology to find RREF of its corresponding augmented matrix.

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(c)

Given this solution set, does [−1−61] belong to span{[10−3],[−1−32]}?
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Observation 2.2.8.

The following are all equivalent statements:

  • The vector b→ belongs to span{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 2.2.9.

Determine if [3−215] belongs to span{[10−32],[−1−322]} by solving an appropriate vector equation.
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Activity 2.2.10.

Determine if [−1−90] belongs to span{[10−3],[−1−32]} by solving an appropriate vector equation.
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Activity 2.2.11.

Does the third-degree polynomial 3y3−2y2+y+5 in P3 belong to span{y3−3y+2,−y3−3y2+2y+2}?

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(a)

Reinterpret this question as a question about the solution(s) of a polynomial equation:

x1(⋯?⋯)+x2(⋯?⋯)=(⋯?⋯)
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(b)

Write a Euclidean vector equation that has the same solution set:

x1[????]+x2[????]=[????]
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(c)

Answer this equivalent question, and use its solution to answer the original question.
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Activity 2.2.12.

Does the polynomial x2+x+1 belong to span{x2−x,x+1,x2−1}?

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Activity 2.2.13.

Does the matrix [3−215] belong to span{[10−32],[−1−322]}?

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(a)

Reinterpret this question as a question about the solution(s) of a matrix equation.

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(b)

Answer this equivalent question, and use its solution to answer the original question.
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Subsection 2.2.2 Videos

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Figure 7. Video: Linear combinations
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Subsection 2.2.3 Slideshow

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Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/VS2.slides.html.

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Exercises 2.2.4 Exercises

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Exercises available at https://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/VS2/.

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Subsection 2.2.5 Mathematical Writing Explorations

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Exploration 2.2.14.

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 2.2.6 Sample Problem and Solution

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Sample problem Example B.1.6.

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