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Section 2.3 Spanning Sets (VS3)

Subsection 2.3.1 Class Activities

Observation 2.3.1.

Any single non-zero vector/number x in R1 spans R1, since R1={cx|c∈R}.

Figure 8. An R1 vector

Activity 2.3.2.

How many vectors are required to span R2? Sketch a drawing in the xy plane to support your answer.

Figure 9. The xy plane R2
  1. 1

  2. 2

  3. 3

  4. 4

  5. Infinitely Many

Activity 2.3.3.

How many vectors are required to span R3?

Figure 10. R3 space
  1. 1

  2. 2

  3. 3

  4. 4

  5. Infinitely Many

Activity 2.3.5.

Choose any vector [???] in R3 that is not in span{[1βˆ’10],[βˆ’201]} by using technology to verify that RREF[1βˆ’2?βˆ’10?01?]=[100010001]. (Why does this work?)

Activity 2.3.7.

Consider the set of vectors S={[230βˆ’1],[1βˆ’430],[17βˆ’3βˆ’1],[0357],[313716]} and the question β€œDoes R4=spanS?”

(a)

Rewrite this question in terms of the solutions to a vector equation.

(b)

Answer your new question, and use this to answer the original question.

Activity 2.3.8.

Consider the set of third-degree polynomials

S={2x3+3x2βˆ’1,2x3+3,3x3+13x2+7x+16,βˆ’x3+10x2+7x+14,4x3+3x2+2}.

and the question β€œDoes P3=spanS?”

(a)

Rewrite this question to be about the solutions to a polynomial equation.

(b)

Answer your new question, and use this to answer the original question.

Activity 2.3.9.

Consider the set of matrices

S={[1301],[1βˆ’110],[1002]}

and the question β€œDoes M2,2=spanS?”

(a)

Rewrite this as a question about the solutions to a matrix equation.

(b)

Answer your new question, and use this to answer the original question.

Activity 2.3.10.

Let vβ†’1,vβ†’2,vβ†’3∈R7 be three vectors, and suppose wβ†’ is another vector with wβ†’βˆˆspan{vβ†’1,vβ†’2,vβ†’3}. What can you conclude about span{wβ†’,vβ†’1,vβ†’2,vβ†’3}?

  1. span{w→,v→1,v→2,v→3} is larger than span{v→1,v→2,v→3}.

  2. span{w→,v→1,v→2,v→3}=span{v→1,v→2,v→3}.

  3. span{w→,v→1,v→2,v→3} is smaller than span{v→1,v→2,v→3}.

Subsection 2.3.2 Videos

Figure 12. Video: Determining if a set spans a Euclidean space

Subsection 2.3.3 Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/VS3.slides.html.

Exercises 2.3.4 Exercises

Exercises available at https://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/VS3/.

Subsection 2.3.5 Mathematical Writing Explorations

Exploration 2.3.11.

Construct each of the following, or show that it is impossible:
  • A set of 2 vectors that spans R3

  • A set of 3 vectors that spans R3

  • A set of 3 vectors that does not span R3

  • A set of 4 vectors that spans R3

For any of the sets you constructed that did span the required space, are any of the vectors a linear combination of the others in your set?

Exploration 2.3.12.

Based on these results, generalize this a conjecture about how a set of nβˆ’1,n and n+1 vectors would or would not span Rn.

Subsection 2.3.6 Sample Problem and Solution

Sample problem Example B.1.7.

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