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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.

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

- ⎡10121−1−18140−2⎦⎥⎤
- ⎡10121−1−181⎦⎥⎤

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*=13*y*−2*z*=−52*x*−2*y*=−6

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

Q1. Given the matrices:

A =

[2−13−1]*A*=[2−13−1], B =

[−25−4−1]*B*=[−25−4−1]

Find A+B*A*+*B*:

*A*+*B*=[04−1−2]*A*+*B*=[04−10]

Q2. Using the same A*A*, B*B* matrices, find A-B*A*−*B*:

*A*−*B*=[04−1−2]*A*−*B*=[4−670]

Q3. Using the same A*A*, B*B* matrices, find 3A3*A*:

- 3
*A*=[323−113−1] - 3
*A*=[6−39−3]

Q4. Using the same A*A*, B*B* matrices, what is the row 1, column 1 entry of the matrix resulting from 3A-B3*A*−*B*?

`Answer: `

Q1. Consider the matrices:

A =

⎡⎣5−27320141⎤⎦

*A*=⎣⎢⎡5−27320141⎦⎥⎤, B =

⎡⎣29−3⎤⎦*B*=⎣⎢⎡29−3⎦⎥⎤

Can we multiply A \cdot B*A*⋅*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 A
*A*and B*B*are not compatible to perform A \cdot B*A*⋅*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 A*A*, B*B* matrices:

Can we multiply B \cdot A*B*⋅*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 A
*A*and B*B*make B \cdot A*B*⋅*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}*D*3*x*1024 and C_{1024×8}*C*1024*x*8, how many columns would the matrix resulting from D \cdot C*D*⋅*C* have?

`Answer: `

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

Q1. Consider the following:

x =

⎡⎣123⎤⎦*x*=⎣⎢⎡123⎦⎥⎤, y =

⎡⎣321⎤⎦*y*=⎣⎢⎡321⎦⎥⎤, z =

⎡⎣048⎤⎦*z*=⎣⎢⎡048⎦⎥⎤

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

- Yes
- No

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⎦⎥

Q1. Find the determinant of the matrix A =

[219−3]*A*=[219−3]

`Answer: `

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]

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⎦⎥⎤

Q1. Compute the dot product of the vectors u =

⎡⎣324⎤⎦*u*=⎣⎢⎡324⎦⎥⎤ and v =

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

`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⎦⎥⎤:

- 17
- 17
- 4
- 15

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

A =

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

⎡⎣⎢⎢−1216⎤⎦⎥⎥*b*=⎣⎢⎢⎢⎡−1216⎦⎥⎥⎥

*x*^=[121]- ^=[−331]
- ^=[221]

Q1. For the matrices A =

[20241−2]*A*=[20241−2], B =

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

What is the number in matrix C*C*‘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:

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

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

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

Q6. Are the vectors u =

⎡⎣05−4⎤⎦*u*=⎣⎢⎡05−4⎦⎥⎤ and v =

⎡⎣15−4−5⎤⎦*v*=⎣⎢⎡15−4−5⎦⎥⎤ orthogonal?

- Yes
- No

Q7. Find the inverse of A*A*, 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=b*Ax*=*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 A*A* and b*b* so they are no longer decimals.

*x*^=[12]- ^=[331]
- ^=[1−1]
- ^=[221]

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