Recall that the dot product of two vectors (1d matrices) produces a scalar value. The dot product is slightly confusing as the scalar value produced can have an arbitrary meaning that simply represents the mathematical operations of multiplying and summing. In other words, the dot product can simply represent the linear projection of vector onto the number line. This interpretation will be used repeatedly throughout machine learning as our main goal will be to take some features dotted with some weights/parameters. Once again, this can be though of as projecting our features onto a number line where the projection acts as our prediction! Keep this idea in mind as it might not make complete sense as of yet. It also important to know that the dot product can additionally take on other meanings such as a geometric meaning which represents how similar any two vectors are when projected onto one another. Meaning, how much one vector points in the direction of another. Given the following two vectors compute the dot product using the following equation < x, w >:= w. Note that the symbols <> and are used to represent the dot product. |1| |1 x=2 w = 1 09 O-1 O 10 2 O
Recall that the dot product of two vectors (1d matrices) produces a scalar value. The dot product is slightly confusing as the scalar value produced can have an arbitrary meaning that simply represents the mathematical operations of multiplying and summing. In other words, the dot product can simply represent the linear projection of vector onto the number line. This interpretation will be used repeatedly throughout machine learning as our main goal will be to take some features dotted with some weights/parameters. Once again, this can be though of as projecting our features onto a number line where the projection acts as our prediction! Keep this idea in mind as it might not make complete sense as of yet. It also important to know that the dot product can additionally take on other meanings such as a geometric meaning which represents how similar any two vectors are when projected onto one another. Meaning, how much one vector points in the direction of another. Given the following two vectors compute the dot product using the following equation < x, w >:= w. Note that the symbols <> and are used to represent the dot product. |1| |1 x=2 w = 1 09 O-1 O 10 2 O
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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![Recall that the dot product of two vectors (1d matrices) produces a scalar value. The dot product is slightly confusing as the scalar value produced can have an arbitrary meaning that simply represents the mathematical operations of multiplying and summing. In other words, the dot product can simply represent the linear projection of vector onto the number line. This interpretation will be used repeatedly throughout machine learning as our main goal will be to take some features dotted with some weights/parameters. Once again, this can be thought of as projecting our features onto a number line where the projection acts as our prediction! Keep this idea in mind as it might not make complete sense as of yet.
It is also important to know that the dot product can additionally take on other meanings such as a geometric meaning which represents how similar any two vectors are when projected onto one another. Meaning, how much one vector points in the direction of another.
**Given the following two vectors compute the dot product using the following equation \( \langle x, w \rangle := x^T w \). Note that the symbols <> and \(-\) are used to represent the dot product.**
\[ x = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, w = \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix} \]
- ○ 9
- ○ -1
- ○ 10
- ○ 2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e96339b-d3d9-4039-844d-3fc091742cc5%2Ff099dae6-ced7-447a-9b77-dd606eda871b%2Faaykq6_processed.png&w=3840&q=75)
Transcribed Image Text:Recall that the dot product of two vectors (1d matrices) produces a scalar value. The dot product is slightly confusing as the scalar value produced can have an arbitrary meaning that simply represents the mathematical operations of multiplying and summing. In other words, the dot product can simply represent the linear projection of vector onto the number line. This interpretation will be used repeatedly throughout machine learning as our main goal will be to take some features dotted with some weights/parameters. Once again, this can be thought of as projecting our features onto a number line where the projection acts as our prediction! Keep this idea in mind as it might not make complete sense as of yet.
It is also important to know that the dot product can additionally take on other meanings such as a geometric meaning which represents how similar any two vectors are when projected onto one another. Meaning, how much one vector points in the direction of another.
**Given the following two vectors compute the dot product using the following equation \( \langle x, w \rangle := x^T w \). Note that the symbols <> and \(-\) are used to represent the dot product.**
\[ x = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, w = \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix} \]
- ○ 9
- ○ -1
- ○ 10
- ○ 2
![Furthermore, we can use the dot product definition
\[ \langle \mathbf{x}, \mathbf{y} \rangle = \|\mathbf{x}\|_2 \|\mathbf{y}\|_2 \cos(\theta) \]
to further determine the angle between two vectors. For example, see the below image for computing and formula for computing the angle between two vectors using the dot product definition.
The diagram shows two vectors, \( \mathbf{x} \) and \( \mathbf{y} \), with an angle \( \theta \) between them. The equation for calculating the angle is given as:
\[ \theta = \arccos \left( \frac{\mathbf{x} \cdot \mathbf{y}}{|\mathbf{x}| |\mathbf{y}|} \right) \]
Machine learning often uses this idea of cosine similarity and the angle between vectors to compute the similarity between different data samples using their feature vectors (i.e., the columns of a dataset)! See examples [here](#).
---
**Given the following two vectors, find the angle between them (in degrees). Use the above equation, in the picture, but use the L2 norm or \( \| \cdot \|_2 \) when computing \( |\mathbf{x}| \) and \( |\mathbf{y}| \). Recall to compute the L2-norm for a vector \( \mathbf{x} \), we do the following:**
\[
\|\mathbf{x}\|_2 = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}.
\]
\[
\mathbf{x} =
\begin{bmatrix}
1 \\
1
\end{bmatrix}, \quad \mathbf{y} =
\begin{bmatrix}
-2 \\
-2
\end{bmatrix}.
\]
- [ ] 180
- [ ] -1
- [ ] 1
- [ ] 90](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e96339b-d3d9-4039-844d-3fc091742cc5%2Ff099dae6-ced7-447a-9b77-dd606eda871b%2Fo9av5bv_processed.png&w=3840&q=75)
Transcribed Image Text:Furthermore, we can use the dot product definition
\[ \langle \mathbf{x}, \mathbf{y} \rangle = \|\mathbf{x}\|_2 \|\mathbf{y}\|_2 \cos(\theta) \]
to further determine the angle between two vectors. For example, see the below image for computing and formula for computing the angle between two vectors using the dot product definition.
The diagram shows two vectors, \( \mathbf{x} \) and \( \mathbf{y} \), with an angle \( \theta \) between them. The equation for calculating the angle is given as:
\[ \theta = \arccos \left( \frac{\mathbf{x} \cdot \mathbf{y}}{|\mathbf{x}| |\mathbf{y}|} \right) \]
Machine learning often uses this idea of cosine similarity and the angle between vectors to compute the similarity between different data samples using their feature vectors (i.e., the columns of a dataset)! See examples [here](#).
---
**Given the following two vectors, find the angle between them (in degrees). Use the above equation, in the picture, but use the L2 norm or \( \| \cdot \|_2 \) when computing \( |\mathbf{x}| \) and \( |\mathbf{y}| \). Recall to compute the L2-norm for a vector \( \mathbf{x} \), we do the following:**
\[
\|\mathbf{x}\|_2 = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}.
\]
\[
\mathbf{x} =
\begin{bmatrix}
1 \\
1
\end{bmatrix}, \quad \mathbf{y} =
\begin{bmatrix}
-2 \\
-2
\end{bmatrix}.
\]
- [ ] 180
- [ ] -1
- [ ] 1
- [ ] 90
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