Linear Algebra Quiz

1. What is the magnitude of vector:

\vec{v} = [3,4]

2. What does the following code output?

3. Which pair of vectors is directionally equivalent (have the same direction)?

\vec{a} = [2,4],\ \vec{b} = [1, 2]

4. 5. Which transformation matrix scales $x$ by 3 and $y$ by 0.5?

6. The following Python code represents a scaling transformation that scales $x$ by 3 and $y$ by 0.5.

What is the output?

7. A rotation matrix rotates vectors counterclockwise around the origin by a given angle $\theta$ .

The general 2D rotation matrix is given by:

R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & cos(\theta) \end{bmatrix}

For $\theta = 90$ (counterclockwise rotation), we substitute:

R(90\degree) = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

Which matrix correctly represents this transformation?

8. What is printed?

9. Which LU decomposition correctly factors the matrix $A$ into $L$ and $U$ ?

A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}

10. Which matrices $L$ and $U$ will be in the output?

11. Which $Q$ matrix is correct for the QR decomposition of $\begin{bmatrix}3&4\\0&5\end{bmatrix}$ ?

12. Which matrix $Q$ is correct for the QR decomposition of $A$ in code?

13. An orthonormal matrix always has eigenvalues that are:

14. When an eigenvector satisfies the equation $A\vec{x} = \lambda\vec{x}$ ?

15. What does this print?

16. Which statement is true?

What is the magnitude of vector:

\vec{v} = [3,4]

Select the correct answer

5.5

What does the following code output?

Select the correct answer

[4, 2]

[2, 1]

[3, 0]

[1, 1]

Which pair of vectors is directionally equivalent (have the same direction)?

\vec{a} = [2,4],\ \vec{b} = [1, 2]

Select the correct answer

They are not equivalent.

Only if scaled by −1.

Yes, they point in the same direction.

No, different magnitudes.

A zero vector has components:

[

]

Which transformation matrix scales $x$ by 3 and $y$ by 0.5?

Select the correct answer

\begin{bmatrix} 3 & 0 \\ 0 & 0.5 \end{bmatrix}

\begin{bmatrix} 0.5 & 0 \\ 0 & 3 \end{bmatrix}

\begin{bmatrix} 3 & 0.5 \\ 0 & 1 \end{bmatrix}

\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

The following Python code represents a scaling transformation that scales $x$ by 3 and $y$ by 0.5.

What is the output?

Select the correct answer

\begin{bmatrix} 6 \\ 2 \end{bmatrix}

\begin{bmatrix} 2 \\ 4 \end{bmatrix}

\begin{bmatrix} 3 \\ 0.5 \end{bmatrix}

\begin{bmatrix} 8 \\ 0 \end{bmatrix}

A rotation matrix rotates vectors counterclockwise around the origin by a given angle $\theta$ .
The general 2D rotation matrix is given by:

R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & cos(\theta) \end{bmatrix}

For $\theta = 90$ (counterclockwise rotation), we substitute:

R(90\degree) = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

Which matrix correctly represents this transformation?

Select the correct answer

\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}

\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

\begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}

What is printed?

Select the correct answer

[0, 2]

[-2, 0]

[0, -2]

[2, 2]

Which LU decomposition correctly factors the matrix $A$ into $L$ and $U$ ?

A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}

Select the correct answer

\begin{bmatrix} 1 & 0 \\ 3 & 1 \end{bmatrix} \begin{bmatrix} 2 & 4 \\ 0 & -4 \end{bmatrix}

\begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 2 & 4 \\ 0 & 4 \end{bmatrix}

\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}

\begin{bmatrix} 1 & 0 \\ 0.5 & 1 \end{bmatrix} \begin{bmatrix} 2 & 4 \\ 0 & 6 \end{bmatrix}

Which matrices $L$ and $U$ will be in the output?

Select the correct answer

L = \begin{bmatrix} 1 & 0 \\ 0.5 & 1 \end{bmatrix},\ \ U = \begin{bmatrix} 4 & 3 \\ 0 & 1.5 \end{bmatrix}

L = \begin{bmatrix} 1 & 0 \\ 0.5 & 1 \end{bmatrix},\ \ U = \begin{bmatrix} 6 & 3 \\ 0 & 1.5 \end{bmatrix}

L = \begin{bmatrix} 1 & 0 \\ 0.5 & 1 \end{bmatrix},\ \ U = \begin{bmatrix} 4 & 3 \\ 0 & 3 \end{bmatrix}

L = \begin{bmatrix} 1 & 0 \\ 0.5 & 1 \end{bmatrix},\ \ U = \begin{bmatrix} 6 & 3 \\ 0 & 3 \end{bmatrix}

Which $Q$ matrix is correct for the QR decomposition of $\begin{bmatrix}3&4\\0&5\end{bmatrix}$ ?

Select the correct answer

\begin{bmatrix} 0.6&0.8 \\ 0&1 \end{bmatrix}

\begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix}

\begin{bmatrix} 3&4 \\ 0&5 \end{bmatrix}

\begin{bmatrix} 0.3&0.4 \\ 0&0.5 \end{bmatrix}

Which matrix $Q$ is correct for the QR decomposition of $A$ in code?

Select the correct answer

Q = \begin{bmatrix} -0.316 & -0.949 \\ -0.949 & 0.316 \end{bmatrix}

Q = \begin{bmatrix} -0.316 & 0.949 \\ -0.949 & -0.316 \end{bmatrix}

Q = \begin{bmatrix} 0.316 & -0.949 \\ 0.949 & 0.316 \end{bmatrix}

Q = \begin{bmatrix} 0.316 & 0.949 \\ 0.949 & -0.316 \end{bmatrix}

An orthonormal matrix always has eigenvalues that are:

Select the correct answer

Positive real numbers

Negative real numbers

Either 1 or -1

Zero or complex

When an eigenvector satisfies the equation $A\vec{x} = \lambda\vec{x}$ ?

Select the correct answer

If $\lambda$ is a matrix

If $\vec{x}$ is a scalar

If $\vec{x}$ is a nonzero vector

If $A$ is a scalar

What does this print?

Select the correct answer

[3, 1]

[1, 3]

[2, 2]

[0, 4]

Which statement is true?

Select the correct answer

All vectors are eigenvectors.

Eigenvectors always have eigenvalue 1.

Eigenvectors maintain their direction after transformation.

Eigenvalues determine the rotation angle.

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Como podemos melhorá-lo?

Obrigado pelo seu feedback!

Seção 4. Capítulo 13

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Linear Algebra Quiz

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1. What is the magnitude of vector:

\vec{v} = [3,4]

2. What does the following code output?

3. Which pair of vectors is directionally equivalent (have the same direction)?

\vec{a} = [2,4],\ \vec{b} = [1, 2]

4. 5. Which transformation matrix scales $x$ by 3 and $y$ by 0.5?

6. The following Python code represents a scaling transformation that scales $x$ by 3 and $y$ by 0.5.

What is the output?

7. A rotation matrix rotates vectors counterclockwise around the origin by a given angle $\theta$ .

The general 2D rotation matrix is given by:

R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & cos(\theta) \end{bmatrix}

For $\theta = 90$ (counterclockwise rotation), we substitute:

R(90\degree) = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

Which matrix correctly represents this transformation?

8. What is printed?

9. Which LU decomposition correctly factors the matrix $A$ into $L$ and $U$ ?

A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}

10. Which matrices $L$ and $U$ will be in the output?

11. Which $Q$ matrix is correct for the QR decomposition of $\begin{bmatrix}3&4\\0&5\end{bmatrix}$ ?

12. Which matrix $Q$ is correct for the QR decomposition of $A$ in code?

13. An orthonormal matrix always has eigenvalues that are:

14. When an eigenvector satisfies the equation $A\vec{x} = \lambda\vec{x}$ ?

15. What does this print?

16. Which statement is true?

What is the magnitude of vector:

\vec{v} = [3,4]

Select the correct answer

5.5

What does the following code output?

Select the correct answer

[4, 2]

[2, 1]

[3, 0]

[1, 1]

Which pair of vectors is directionally equivalent (have the same direction)?

\vec{a} = [2,4],\ \vec{b} = [1, 2]

Select the correct answer

They are not equivalent.

Only if scaled by −1.

Yes, they point in the same direction.

No, different magnitudes.

A zero vector has components:

[

]

Which transformation matrix scales $x$ by 3 and $y$ by 0.5?

Select the correct answer

\begin{bmatrix} 3 & 0 \\ 0 & 0.5 \end{bmatrix}

\begin{bmatrix} 0.5 & 0 \\ 0 & 3 \end{bmatrix}

\begin{bmatrix} 3 & 0.5 \\ 0 & 1 \end{bmatrix}

\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

The following Python code represents a scaling transformation that scales $x$ by 3 and $y$ by 0.5.

What is the output?

Select the correct answer

\begin{bmatrix} 6 \\ 2 \end{bmatrix}

\begin{bmatrix} 2 \\ 4 \end{bmatrix}

\begin{bmatrix} 3 \\ 0.5 \end{bmatrix}

\begin{bmatrix} 8 \\ 0 \end{bmatrix}

A rotation matrix rotates vectors counterclockwise around the origin by a given angle $\theta$ .
The general 2D rotation matrix is given by:

R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & cos(\theta) \end{bmatrix}

For $\theta = 90$ (counterclockwise rotation), we substitute:

R(90\degree) = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

Which matrix correctly represents this transformation?

Select the correct answer

\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}

\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}

\begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}

What is printed?

Select the correct answer

[0, 2]

[-2, 0]

[0, -2]

[2, 2]

Which LU decomposition correctly factors the matrix $A$ into $L$ and $U$ ?

A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}

Select the correct answer

\begin{bmatrix} 1 & 0 \\ 3 & 1 \end{bmatrix} \begin{bmatrix} 2 & 4 \\ 0 & -4 \end{bmatrix}

\begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 2 & 4 \\ 0 & 4 \end{bmatrix}

\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}

\begin{bmatrix} 1 & 0 \\ 0.5 & 1 \end{bmatrix} \begin{bmatrix} 2 & 4 \\ 0 & 6 \end{bmatrix}

Which matrices $L$ and $U$ will be in the output?

Select the correct answer

L = \begin{bmatrix} 1 & 0 \\ 0.5 & 1 \end{bmatrix},\ \ U = \begin{bmatrix} 4 & 3 \\ 0 & 1.5 \end{bmatrix}

L = \begin{bmatrix} 1 & 0 \\ 0.5 & 1 \end{bmatrix},\ \ U = \begin{bmatrix} 6 & 3 \\ 0 & 1.5 \end{bmatrix}

L = \begin{bmatrix} 1 & 0 \\ 0.5 & 1 \end{bmatrix},\ \ U = \begin{bmatrix} 4 & 3 \\ 0 & 3 \end{bmatrix}

L = \begin{bmatrix} 1 & 0 \\ 0.5 & 1 \end{bmatrix},\ \ U = \begin{bmatrix} 6 & 3 \\ 0 & 3 \end{bmatrix}

Which $Q$ matrix is correct for the QR decomposition of $\begin{bmatrix}3&4\\0&5\end{bmatrix}$ ?

Select the correct answer

\begin{bmatrix} 0.6&0.8 \\ 0&1 \end{bmatrix}

\begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix}

\begin{bmatrix} 3&4 \\ 0&5 \end{bmatrix}

\begin{bmatrix} 0.3&0.4 \\ 0&0.5 \end{bmatrix}

Which matrix $Q$ is correct for the QR decomposition of $A$ in code?

Select the correct answer

Q = \begin{bmatrix} -0.316 & -0.949 \\ -0.949 & 0.316 \end{bmatrix}

Q = \begin{bmatrix} -0.316 & 0.949 \\ -0.949 & -0.316 \end{bmatrix}

Q = \begin{bmatrix} 0.316 & -0.949 \\ 0.949 & 0.316 \end{bmatrix}

Q = \begin{bmatrix} 0.316 & 0.949 \\ 0.949 & -0.316 \end{bmatrix}

An orthonormal matrix always has eigenvalues that are:

Select the correct answer

Positive real numbers

Negative real numbers

Either 1 or -1

Zero or complex

When an eigenvector satisfies the equation $A\vec{x} = \lambda\vec{x}$ ?

Select the correct answer

If $\lambda$ is a matrix

If $\vec{x}$ is a scalar

If $\vec{x}$ is a nonzero vector

If $A$ is a scalar

What does this print?

Select the correct answer

[3, 1]

[1, 3]

[2, 2]

[0, 4]

Which statement is true?

Select the correct answer

All vectors are eigenvectors.

Eigenvectors always have eigenvalue 1.

Eigenvectors maintain their direction after transformation.

Eigenvalues determine the rotation angle.

Tudo estava claro?

Como podemos melhorá-lo?

Obrigado pelo seu feedback!

Seção 4. Capítulo 13

Linear Algebra Quiz

1. What is the magnitude of vector:

2. What does the following code output?

3. Which pair of vectors is directionally equivalent (have the same direction)?

4.

5. Which transformation matrix scales xxx by 3 and yyy by 0.5?

6. The following Python code represents a scaling transformation that scales xxx by 3 and yyy by 0.5.

7. A rotation matrix rotates vectors counterclockwise around the origin by a given angle θ\thetaθ.

8. What is printed?

9. Which LU decomposition correctly factors the matrix AAA into LLL and UUU?

10. Which matrices LLL and UUU will be in the output?

11. Which QQQ matrix is correct for the QR decomposition of [3405]\begin{bmatrix}3&4\\0&5\end{bmatrix}[30​45​]?

12. Which matrix QQQ is correct for the QR decomposition of AAA in code?

13. An orthonormal matrix always has eigenvalues that are:

14. When an eigenvector satisfies the equation Ax⃗=λx⃗A\vec{x} = \lambda\vec{x}Ax=λx?

15. What does this print?

16. Which statement is true?

Awesome!

Linear Algebra Quiz

1. What is the magnitude of vector:

2. What does the following code output?

3. Which pair of vectors is directionally equivalent (have the same direction)?

4.

5. Which transformation matrix scales xxx by 3 and yyy by 0.5?

6. The following Python code represents a scaling transformation that scales xxx by 3 and yyy by 0.5.

7. A rotation matrix rotates vectors counterclockwise around the origin by a given angle θ\thetaθ.

8. What is printed?

9. Which LU decomposition correctly factors the matrix AAA into LLL and UUU?

10. Which matrices LLL and UUU will be in the output?

11. Which QQQ matrix is correct for the QR decomposition of [3405]\begin{bmatrix}3&4\\0&5\end{bmatrix}[30​45​]?

12. Which matrix QQQ is correct for the QR decomposition of AAA in code?

13. An orthonormal matrix always has eigenvalues that are:

14. When an eigenvector satisfies the equation Ax⃗=λx⃗A\vec{x} = \lambda\vec{x}Ax=λx?

15. What does this print?

16. Which statement is true?

5. Which transformation matrix scales $x$ by 3 and $y$ by 0.5?

6. The following Python code represents a scaling transformation that scales $x$ by 3 and $y$ by 0.5.

7. A rotation matrix rotates vectors counterclockwise around the origin by a given angle $\theta$ .

9. Which LU decomposition correctly factors the matrix $A$ into $L$ and $U$ ?

10. Which matrices $L$ and $U$ will be in the output?

11. Which $Q$ matrix is correct for the QR decomposition of $\begin{bmatrix}3&4\\0&5\end{bmatrix}$ ?

12. Which matrix $Q$ is correct for the QR decomposition of $A$ in code?

14. When an eigenvector satisfies the equation $A\vec{x} = \lambda\vec{x}$ ?

5. Which transformation matrix scales $x$ by 3 and $y$ by 0.5?

6. The following Python code represents a scaling transformation that scales $x$ by 3 and $y$ by 0.5.

7. A rotation matrix rotates vectors counterclockwise around the origin by a given angle $\theta$ .

9. Which LU decomposition correctly factors the matrix $A$ into $L$ and $U$ ?

10. Which matrices $L$ and $U$ will be in the output?

11. Which $Q$ matrix is correct for the QR decomposition of $\begin{bmatrix}3&4\\0&5\end{bmatrix}$ ?

12. Which matrix $Q$ is correct for the QR decomposition of $A$ in code?

14. When an eigenvector satisfies the equation $A\vec{x} = \lambda\vec{x}$ ?