Consider the vectors V1, V2, V3, and b. V1 4 V2 = 5, V3 6 ,b = 7
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![### Vectors in Linear Algebra
Consider the vectors **v₁**, **v₂**, **v₃**, and **b** given below:
\[
v₁ = \begin{bmatrix}
1 \\
4 \\
6
\end{bmatrix}, \quad v₂ = \begin{bmatrix}
2 \\
5 \\
7
\end{bmatrix}, \quad v₃ = \begin{bmatrix}
3 \\
6 \\
8
\end{bmatrix}, \quad b = \begin{bmatrix}
4 \\
7 \\
9
\end{bmatrix}
\]
In these vectors:
- **v₁** is a 3-dimensional vector with elements 1, 4, and 6.
- **v₂** is a 3-dimensional vector with elements 2, 5, and 7.
- **v₃** is a 3-dimensional vector with elements 3, 6, and 8.
- **b** is a 3-dimensional vector with elements 4, 7, and 9.
These vectors can be used in various linear algebra operations such as vector addition, scalar multiplication, dot product, and cross product, depending on the context and the dimensions involved. They can also be part of systems of linear equations, matrix operations, and transformations in higher-dimensional space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee22cf2f-b974-4b00-a3cf-09b388e7d65d%2Fb4f9405c-82e6-4580-9a73-f0233e78f2ac%2F4lsa6g_processed.jpeg&w=3840&q=75)
![**Question:**
Is the set of vectors S = {v₁, v₂, v₃} linearly independent? Explain.
---
**Explanation:**
To determine whether the set of vectors \( S = \{v_1, v_2, v_3\} \) is linearly independent, we need to assess whether the only solution to the equation
\[ c_1 v_1 + c_2 v_2 + c_3 v_3 = 0 \]
is the trivial solution where \( c_1 = 0 \), \( c_2 = 0 \), and \( c_3 = 0 \).
1. Formulate the vectors in matrix form.
2. Perform row-reduction (Gaussian elimination) on the matrix formed by these vectors.
3. If, after row-reduction, the matrix has a pivot (leading 1) in every column, then the vectors are linearly independent.
4. If any column does not have a pivot, the vectors are linearly dependent.
A step-by-step matrix representation and row-reduction would clearly demonstrate the linear dependence or independence of the vectors. Ensure to provide the vectors' coordinates or their explicit representations for a detailed verification.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee22cf2f-b974-4b00-a3cf-09b388e7d65d%2Fb4f9405c-82e6-4580-9a73-f0233e78f2ac%2Fsi6ulln_processed.jpeg&w=3840&q=75)
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