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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+2y−z=4y+8z=0x−y+z=−2
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
Q1. Given the matrices:
A =
[2−13−1]A=[2−13−1], B =
[−25−4−1]B=[−25−4−1]
Find A+BA+B:
Q2. Using the same AA, BB matrices, find A-BA−B:
Q3. Using the same AA, BB matrices, find 3A3A:
Q4. Using the same AA, BB matrices, what is the row 1, column 1 entry of the matrix resulting from 3A-B3A−B?
Answer:
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.
Q2. Consider the same AA, BB matrices:
Can we multiply B \cdot AB⋅A? If so, compute the resulting matrix.
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:
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.
Q2. Determine if the following statement is true or false:
A vector is not a matrix.
Q1. Consider the following:
x =
⎡⎣123⎤⎦x=⎣⎢⎡123⎦⎥⎤, y =
⎡⎣321⎤⎦y=⎣⎢⎡321⎦⎥⎤, z =
⎡⎣048⎤⎦z=⎣⎢⎡048⎦⎥⎤
Are {x,y,zx,y,z} linearly independent?
Q1. Find the inverse of A, where A =
⎡⎣70−32341−1−2⎤⎦A=⎣⎢⎡70−32341−1−2⎦⎥
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:
Q1. Find the eigenvalues of the matrix:
A =
⎡⎣52−3254006⎤⎦A=⎣⎢⎡52−3254006⎦⎥⎤
Answer:
Q2. Find the corresponding eigenvectors from Question 1:
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⎦⎥⎤:
Q1. Find the least squares solution to Ax=bAx=b using :
A =
⎡⎣⎢⎢1111−6−217⎤⎦⎥⎥A=⎣⎢⎢⎢⎡1111−6−217⎦⎥⎥⎥⎤ and b =
⎡⎣⎢⎢−1216⎤⎦⎥⎥b=⎣⎢⎢⎢⎡−1216⎦⎥⎥⎥
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⎦⎥⎤:
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.
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.
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.
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