Essential Linear Algebra for Data Science Coursera Quiz Answers

Get Essential Linear Algebra for Data Science Coursera Quiz Answers

Are you interested in Data Science but lack the math background for it? Has math always been a tough subject that you tend to avoid? This course will teach you the most fundamental Linear Algebra that you will need for a career in Data Science without a ton of unnecessary proofs and concepts that you may never use. Consider this an expressway to Data Science with approachable methods and friendly concepts that will guide you to truly understanding the most important ideas in Linear Algebra.

This course is designed to prepare learners to successfully complete Statistical Modeling for Data Science Application, which is part of CU Boulder’s Master of Science in Data Science (MS-DS) program.

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Week 1: Linear Systems and Gaussian Elimination

Quiz 1: Practice – Linear System -> Matrix Format

Q1. Transfer the following linear system into matrix notation: x+2y−z=4y+8z=0x−y+z=−2

⎡10121−1−18140−2⎦⎥⎤
⎡10121−1−181⎦⎥⎤

Quiz 2: LS -> Matrix + G.E. Full Question Quiz

Q1. Transfer the linear system into matrix notation and apply Gaussian Elimination. Does it have no solutions, one solution, or infinite solutions? What is the solution (unless no solution exists) x−y+z=13y−2z=−52x−2y=−6

Infinite Solutions
x=−2 y=1y=1 z=4z=4
No solution
x=2 y=1y=1 z=-3z=−3

Week 2: Matrix Algebra

Quiz 1: Quiz on Matrix Algebra Sum + Scale

Q1. Given the matrices:

A =

[2−13−1]A=[2−13−1], B =

[−25−4−1]B=[−25−4−1]

Find A+BA+B:

A+B=[04−1−2]
A+B=[04−10]

Q2. Using the same AA, BB matrices, find A-BA−B:

A−B=[04−1−2]
A−B=[4−670]

Q3. Using the same AA, BB matrices, find 3A3A:

3A=[323−113−1]
3A=[6−39−3]

Q4. Using the same AA, BB matrices, what is the row 1, column 1 entry of the matrix resulting from 3A-B3A−B?

Answer:

Quiz 2: Matrix Multiplication

Q1. Consider the matrices:

A =

⎡⎣5−27320141⎤⎦

A=⎣⎢⎡5−27320141⎦⎥⎤, B =

⎡⎣29−3⎤⎦B=⎣⎢⎡29−3⎦⎥⎤

Can we multiply A \cdot BA⋅B? If so, compute the resulting matrix.

Yes, and the resulting matrix is A \cdot B = ⎡⎣34211⎤⎦A⋅B=⎣⎢⎡34211⎦⎥⎤
No, the matrix dimensions of AA and BB are not compatible to perform A \cdot BA⋅B.
Yes, and the resulting matrix is A \cdot B = ⎡⎣10−18−216180236−3⎤⎦A⋅B=⎣⎢⎡10−18−216180236−3⎦⎥
Yes, and the resulting matrix is A \cdot B = ⎡⎣3402⎤⎦A⋅B=⎣⎢⎡3402⎦⎥
Yes, and the resulting matrix is A \cdot B = ⎡⎣37211⎤⎦A⋅B=⎣⎢⎡37211⎦⎥⎤

Q2. Consider the same AA, BB matrices:

Can we multiply B \cdot AB⋅A? If so, compute the resulting matrix.

Yes, and the resulting matrix is B \cdot A = ⎡⎣10−18−216180236−3⎤⎦ B⋅A=⎣⎢⎡10−18−216180236−3⎦⎥⎤
No, the matrix dimensions of AA and BB make B \cdot AB⋅A impossible to compute.
Yes, and the resulting matrix is B \cdot A = ⎡⎣34211⎤⎦B⋅A=⎣⎢⎡34211⎦⎥

Q3. Say we have the matrices D_{3×1024}D3x1024 and C_{1024×8}C1024x8, how many columns would the matrix resulting from D \cdot CD⋅C have?

Answer:

Week 3: Properties of a Linear System

Quiz 1: Why do we use matrices and vectors and not just one?

Q1. Determine if the following statement is true or false:

Vectors are useful as they can show us the direction that something is moving in and have applications to computing things like velocity.

True
False

Q2. Determine if the following statement is true or false:

A vector is not a matrix.

True
False

Quiz 2: Quiz on Linear Independence

Q1. Consider the following:

x =

⎡⎣123⎤⎦x=⎣⎢⎡123⎦⎥⎤, y =

⎡⎣321⎤⎦y=⎣⎢⎡321⎦⎥⎤, z =

⎡⎣048⎤⎦z=⎣⎢⎡048⎦⎥⎤

Are {x,y,zx,y,z} linearly independent?

Quiz 3: Quiz on Transformations and Inverse

Q1. Find the inverse of A, where A =

⎡⎣70−32341−1−2⎤⎦A=⎣⎢⎡70−32341−1−2⎦⎥

A−1=⎣⎢⎡9−646−618−3491735⎦⎥
A−1=⎣⎢⎡6498−5−34−5921⎦⎥⎤
A−1=⎣⎢⎡−2398−11−34−5721⎦⎥
A−1=⎣⎢⎡7698−714−5921⎦⎥

Week 4: Determinant and Eigens

Quiz 1: Find the Determinant of a 2×2 Matrix

Q1. Find the determinant of the matrix A =

[219−3]A=[219−3]

Answer:

Quiz 2: Find Eigenvalue then Eigenvector of a Matrix (2×2)

Q1. Find the eigenvalues of the matrix:

A =

[0−21−3]A=[0−21−3]

Answer:

Q2. Find the corresponding eigenvectors from Question 1:

[11]
[1−2]
[1−1]
[−12]
[−11]

Quiz 3: Find Eigenvalue then Eigenvector of a Matrix (3×3)

Q1. Find the eigenvalues of the matrix:

A =

⎡⎣52−3254006⎤⎦A=⎣⎢⎡52−3254006⎦⎥⎤

Answer:

Q2. Find the corresponding eigenvectors from Question 1:

⎡111⎦⎥⎤
⎡110⎦⎥
⎡2−10⎦⎥⎤
⎡3−37⎦⎥⎤
⎡001⎦⎥⎤
⎡2−37⎦⎥⎤

Week 5: Projections and Least Squares

Quiz 1: Important Final Concepts

Q1. Compute the dot product of the vectors u =

⎡⎣324⎤⎦u=⎣⎢⎡324⎦⎥⎤ and v =

⎡⎣206⎤⎦v=⎣⎢⎡206⎦⎥⎤, recall that u^{T}v=v^{T}uuTv=vTu.

Answer:

Q2. Compute the norm of the vector ||v||∣∣v∣∣, where v =

⎡⎣⎢⎢2222⎤⎦⎥⎥

v=⎣⎢⎢⎢⎡2222⎦⎥⎥⎥⎤:

Answer:

Q3. Find the distance between the vectors u =

⎡⎣324⎤⎦u=⎣⎢⎡324⎦⎥⎤ and v =

⎡⎣152⎤⎦v=⎣⎢⎡152⎦⎥⎤:

Quiz 2: Finding Least Squares Solutions

Q1. Find the least squares solution to Ax=bAx=b using :

A =

⎡⎣⎢⎢1111−6−217⎤⎦⎥⎥A=⎣⎢⎢⎢⎡1111−6−217⎦⎥⎥⎥⎤ and b =

⎡⎣⎢⎢−1216⎤⎦⎥⎥b=⎣⎢⎢⎢⎡−1216⎦⎥⎥⎥

x^=[121]
^=[−331]
^=[221]

Quiz 3: Final Exam

Q1. For the matrices A =

[20241−2]A=[20241−2], B =

⎡⎣7−1−7616⎤⎦B=⎣⎢⎡7−1−7616⎦⎥⎤, find C=ABC=AB.

What is the number in matrix CC‘s row 2, column 2?

Answer:

Q2. Find the determinant of A =

[9−5−110]

A=[9−5−110]:

Answer:

Q3. Find the eigenvalues for

[7034]

[7034]:

Answer:

Q4. Compute the dot product of u =

⎡⎣79−2⎤⎦u=⎣⎢⎡79−2⎦⎥⎤ and v =

⎡⎣10−320⎤⎦v=⎣⎢⎡10−320⎦⎥⎤:

3
-3
10
83

Q5. Using Gaussian Elimination, solve the following system of equations:

3x+4y+4z=36x+8y+4z=62y+2z=4

Using a calculator round any remaining fractions to the nearest hundredth.

x=1.67 y=2y=2 z=0z=0
x=−3 y=-2y=−2 z=0z=0
x=−1.67 y=2y=2 z=0z=0
x=0 y=4y=4 z=4z=4

Q6. Are the vectors u =

⎡⎣05−4⎤⎦u=⎣⎢⎡05−4⎦⎥⎤ and v =

⎡⎣15−4−5⎤⎦v=⎣⎢⎡15−4−5⎦⎥⎤ orthogonal?

Q7. Find the inverse of AA, where A =

⎡⎣103826422⎤⎦A=⎣⎢⎡103826422⎦⎥⎤.

Round any fractions to the nearest hundredth in your solution.

A−1=⎣⎢⎡.5.38−.38−.5−.631.13−.5−.13.13⎦⎥
A−1=⎣⎢⎡−.5.38−.38.5−.631.13.5−.13.13⎦⎥⎤
A−1=⎣⎢⎡−.17.38−.5.25.631.11.35.17.11⎦⎥⎤

Q8. Find the least squares solution to Ax=bAx=b using:

A =

⎡⎣⎢⎢.1.1.1.1−.6−.2.1.7⎤⎦⎥⎥A=⎣⎢⎢⎢⎡.1.1.1.1−.6−.2.1.7⎦⎥⎥⎥⎤ and b =

⎡⎣⎢⎢−.1.2.1.6⎤⎦⎥⎥b=⎣⎢⎢⎢⎡−.1.2.1.6⎦⎥⎥⎥⎤

Hint: This problem can be made easier by scaling AA and bb so they are no longer decimals.

x^=[12]
^=[331]
^=[1−1]
^=[221]

Conclusion:

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This Problem is intended for audiences of all experiences who are interested in learning about Data Science in a business context; there are no prerequisites.

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