Colorado College Linear Algebra Matrix & Vectors Questions this assignment is major one, it has two sections, one the problems, second: the reflection pape

Colorado College Linear Algebra Matrix & Vectors Questions this assignment is major one, it has two sections, one the problems, second: the reflection paper you have 5 option to choose from. chose one and write about what’s been asked. M 221
Summer 2020
Unit 2 Assessment
Name:
Definitions
A set of vectors are linearly dependent if and only if the equation
1 1 + 2 2 + ⋯ + = 0 has at least one non-trivial solution.
A set of vectors are linearly independent if and only if the equation
1 1 + 2 2 + ⋯ + = 0 can only be solved by the trivial solution.
The span of a set of vectors is the collection of all possible linear combinations of those vectors.
A matrix is said to be nonsingular if it has an inverse.
A set of vectors forms a basis of Rn if and only if the set spans all of Rn.
A transformation : → is a linear transformation if and only if it satisfies:
1) ( ⃑ + ⃑) = ( ⃑) + ( ⃑) for all ⃑, ⃑ ∈
2) ( ⃑) = ( ⃑), for all ⃑ ∈ , where ∈
Directions: Show all work using correct notation and complete justifications for full credit.
Justifications can be either completed work or written explanations of your thinking. We are
more interested in your explanations and ability to communicate your thinking than concerned
about obtaining correct answers. For problems requesting proofs, communicating your thinking
may take the form of outlining your proof idea. Partial credit will be awarded based on the 4point rubric found on the syllabus. Let either Emmanuel or DW know if you have any questions.
Collaboration with other students, textbooks, notes, and other online resources ARE
PERMITTED for completing this assessment. If you use outside resources make sure to do two
things:
1. Give credit to the source(s) used. For example a statement such as the following would be
a way to give credit to a classmate working with you,
“Awesome Classmate’s Name worked with me on this problem. Their understanding of
as supported my work on this problem.”
2. Write up the solutions to problems on your own. Do not submit the exact same work as
your collaborators or use the same examples you see in online video resources. Show that
your work is indicative of your thinking.
If you have questions about citing your sources, please ask.
Submit your completed assessment as a single PDF to D2L by noon Tuesday, June 16.
1. Explain why any set containing the zero vector (of appropriate size) is not a basis for Rn.
−2 0
2. Suppose that : 2 → 3 is defined by ( ⃑) = ⃑, where = [ 0 0].
0 1
a. What are the domain and codomain of T? Discuss what this means about inputs and
outputs for this transformation.
b. For this transformation, can you find two different input vectors that will give the same
output vector? If so, given an example. If not, why do you think it is not possible?
c. Will this transformation obtain any output vector (in its codomain)? If so, explain why. If
not, give an example vector in the codomain that cannot be obtained by this
transformation and explain why.
3. Vectors are orthogonal if and only if their dot product is 0.
a. Prove that the zero vector in Rn, ⃑⃑⃑⃑⃑
0 is orthogonal to all vectors in Rn.
b. Another method for finding a dot product is ∙ = ‖ ‖‖ ‖cos ( ), where measures
the angle between the two vectors (in the positive direction). Explain how this method for
finding dot products could also be used to complete the proof from part a.
c. Use this alternative understanding of dot products to explain what orthogonal means
geometrically. Assume neither v nor w are the zero-vector.
2
4. Consider the matrix equation [
5
4 7
24
] [ ] = [ ], which is of the form ⃑ = ⃑⃑.
2 2.5
40
a. Identify the solution to this equation.
b. State the corresponding vector equation and interpret the meaning of the solution. How is
the solution represented graphically?
c. State the corresponding system of linear equations and interpret the meaning of the
solution. How is the solution represented graphically?
d. Discuss the meaning of the solution in context of transformation matrices.
1
−3
3
15
5. Solve the equation 1 [−1] + 2 [ 1 ] + 3 [−7] = [−1]. If not possible, explain why.
6
−1
5
16
2
0
6
0
1
6. The augmented matrix [ 1
1 −2 |0] is row equivalent to [0
−3 −1 −4 0
0
0 3
0
1 −5 |0]. Give
0 0
0
four statements about this situation that must be true. Justify each statement.
2 1 − 2
7. Determine whether : 2 → 3 defined by ( ⃑) = [ 1 (1 − 2 )] is a linear transformation. If
3 2
so, prove it. If not, provide a counterexample.
8. Find values of k such that the matrix [
−3 5
] is nonsingular. Explain what such values
−2
2
−3
for k would imply about equations of the form [
−2
also described these equations as ⃑ = ⃑⃑.

5 1
] [ ] = [ 1 ]. In class we have

2 2
2
Reflection
Select one of the following prompts/questions to address. You will not be evaluated for grammar
or spelling.
Option 1: Take one homework problem or task you have worked on this unit that you struggled
to understand and solve, and explain how the struggle itself was valuable. In the context of this
question, describe the struggle and how you overcame the struggle. You might also discuss
whether struggling built aspects of character in you (e.g., endurance, self-confidence,
competence to solve new problems, etc.) and how these virtues might benefit you in later
ventures.
Option 2: What mathematical ideas are you curious to know more about as a result of unit 2?
Give one example of a question about the material that you’d like to explore further, and
describe why this is an interesting question to you.
Option 3: How has your mathematical imagination been enhanced as a result of this unit? Give at
least three examples.
Option 4: Consider one mathematical idea from the unit that you have found beautiful, and
explain why it is beautiful to you. Your answer should: (1) explain the idea in a way that could
be understood by a peer who has taken Calculus I and II but has not yet taken this class, and (2)
address how this beauty is similar to or different from other kinds of beauty that human beings
encounter.
Option 5: Give one example of a mathematical idea from this unit that you found creative, and
explain what you find creative about it. For example, you can choose an instance of creativity
you experienced in your own problem-solving, or something you witnessed in another person’s
thinking or reasoning.
Option 6: As a global community, humans have been plagued by systematic injustice and
discrimination. Either describe how you could use mathematics (possibly topics from this
course) to combat social injustice and promote cultural change, OR describe a situation in which
your mathematics learning experiences have left you feeling personally discriminated against. If
you choose the second path, how did you overcome such injustice and persist with studying
mathematics?
M 221
Summer 2020
Unit 2 Assessment
Name:
Definitions
A set of vectors are linearly dependent if and only if the equation
1 1 + 2 2 + ⋯ + = 0 has at least one non-trivial solution.
A set of vectors are linearly independent if and only if the equation
1 1 + 2 2 + ⋯ + = 0 can only be solved by the trivial solution.
The span of a set of vectors is the collection of all possible linear combinations of those vectors.
A matrix is said to be nonsingular if it has an inverse.
A set of vectors forms a basis of Rn if and only if the set spans all of Rn.
A transformation : → is a linear transformation if and only if it satisfies:
1) ( ⃑ + ⃑) = ( ⃑) + ( ⃑) for all ⃑, ⃑ ∈
2) ( ⃑) = ( ⃑), for all ⃑ ∈ , where ∈
Directions: Show all work using correct notation and complete justifications for full credit.
Justifications can be either completed work or written explanations of your thinking. We are
more interested in your explanations and ability to communicate your thinking than concerned
about obtaining correct answers. For problems requesting proofs, communicating your thinking
may take the form of outlining your proof idea. Partial credit will be awarded based on the 4point rubric found on the syllabus. Let either Emmanuel or DW know if you have any questions.
Collaboration with other students, textbooks, notes, and other online resources ARE
PERMITTED for completing this assessment. If you use outside resources make sure to do two
things:
1. Give credit to the source(s) used. For example a statement such as the following would be
a way to give credit to a classmate working with you,
“Awesome Classmate’s Name worked with me on this problem. Their understanding of
as supported my work on this problem.”
2. Write up the solutions to problems on your own. Do not submit the exact same work as
your collaborators or use the same examples you see in online video resources. Show that
your work is indicative of your thinking.
If you have questions about citing your sources, please ask.
Submit your completed assessment as a single PDF to D2L by noon Tuesday, June 16.
1. Explain why any set containing the zero vector (of appropriate size) is not a basis for Rn.
−2 0
2. Suppose that : 2 → 3 is defined by ( ⃑) = ⃑, where = [ 0 0].
0 1
a. What are the domain and codomain of T? Discuss what this means about inputs and
outputs for this transformation.
b. For this transformation, can you find two different input vectors that will give the same
output vector? If so, given an example. If not, why do you think it is not possible?
c. Will this transformation obtain any output vector (in its codomain)? If so, explain why. If
not, give an example vector in the codomain that cannot be obtained by this
transformation and explain why.
3. Vectors are orthogonal if and only if their dot product is 0.
a. Prove that the zero vector in Rn, ⃑⃑⃑⃑⃑
0 is orthogonal to all vectors in Rn.
b. Another method for finding a dot product is ∙ = ‖ ‖‖ ‖cos ( ), where measures
the angle between the two vectors (in the positive direction). Explain how this method for
finding dot products could also be used to complete the proof from part a.
c. Use this alternative understanding of dot products to explain what orthogonal means
geometrically. Assume neither v nor w are the zero-vector.
2
4. Consider the matrix equation [
5
4 7
24
] [ ] = [ ], which is of the form ⃑ = ⃑⃑.
2 2.5
40
a. Identify the solution to this equation.
b. State the corresponding vector equation and interpret the meaning of the solution. How is
the solution represented graphically?
c. State the corresponding system of linear equations and interpret the meaning of the
solution. How is the solution represented graphically?
d. Discuss the meaning of the solution in context of transformation matrices.
1
−3
3
15
5. Solve the equation 1 [−1] + 2 [ 1 ] + 3 [−7] = [−1]. If not possible, explain why.
6
−1
5
16
2
0
6
0
1
6. The augmented matrix [ 1
1 −2 |0] is row equivalent to [0
−3 −1 −4 0
0
0 3
0
1 −5 |0]. Give
0 0
0
four statements about this situation that must be true. Justify each statement.
2 1 − 2
7. Determine whether : 2 → 3 defined by ( ⃑) = [ 1 (1 − 2 )] is a linear transformation. If
3 2
so, prove it. If not, provide a counterexample.
8. Find values of k such that the matrix [
−3 5
] is nonsingular. Explain what such values
−2
2
−3
for k would imply about equations of the form [
−2
also described these equations as ⃑ = ⃑⃑.

5 1
] [ ] = [ 1 ]. In class we have

2 2
2
Reflection
Select one of the following prompts/questions to address. You will not be evaluated for grammar
or spelling.
Option 1: Take one homework problem or task you have worked on this unit that you struggled
to understand and solve, and explain how the struggle itself was valuable. In the context of this
question, describe the struggle and how you overcame the struggle. You might also discuss
whether struggling built aspects of character in you (e.g., endurance, self-confidence,
competence to solve new problems, etc.) and how these virtues might benefit you in later
ventures.
Option 2: What mathematical ideas are you curious to know more about as a result of unit 2?
Give one example of a question about the material that you’d like to explore further, and
describe why this is an interesting question to you.
Option 3: How has your mathematical imagination been enhanced as a result of this unit? Give at
least three examples.
Option 4: Consider one mathematical idea from the unit that you have found beautiful, and
explain why it is beautiful to you. Your answer should: (1) explain the idea in a way that could
be understood by a peer who has taken Calculus I and II but has not yet taken this class, and (2)
address how this beauty is similar to or different from other kinds of beauty that human beings
encounter.
Option 5: Give one example of a mathematical idea from this unit that you found creative, and
explain what you find creative about it. For example, you can choose an instance of creativity
you experienced in your own problem-solving, or something you witnessed in another person’s
thinking or reasoning.
Option 6: As a global community, humans have been plagued by systematic injustice and
discrimination. Either describe how you could use mathematics (possibly topics from this
course) to combat social injustice and promote cultural change, OR describe a situation in which
your mathematics learning experiences have left you feeling personally discriminated against. If
you choose the second path, how did you overcome such injustice and persist with studying
mathematics?

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