Linear Combination - missing data ?

• MHB
• Yankel
In summary, the question is asking to find the value of k for which the vector ku-3v is a solution to a non-homogeneous system of equations. The correct solution is k=4, which can be determined by setting two equations equal to each other and solving for k.
Yankel
Dear all,

I am trying to solve a question, and I think that something is missing.

It is given that the vectors u and v are solutions to the non-homogeneous system of equations Ax=b.

If the vector ku-3v is a solution to the same system, then:

a) k = 4
b) k = 3
c) k = 0

The correct solution is apparently k = 4. How can you tell ?? I can't figure it out.

Yankel said:
Dear all,

I am trying to solve a question, and I think that something is missing.

It is given that the vectors u and v are solutions to the non-homogeneous system of equations Ax=b.

If the vector ku-3v is a solution to the same system, then:

a) k = 4
b) k = 3
c) k = 0

The correct solution is apparently k = 4. How can you tell ?? I can't figure it out.

Hi Yankel,

You have:
$$A(ku-3v) = kAu - 3Av = (k-3)b$$
On the other hand, as $ku-3v$ is also a solution of the system, you have:
$$A(ku-3v) = b$$
Since $b\ne0$ by hypothesis, the conclusion follows.

So are you saying that k can't be 3 ?

Yankel said:
So are you saying that k can't be 3 ?
You must have $(k-3)b = b$; since $b\ne0$, this implies $k-3 = 1$ and $k=4$. This is the solution you mentioned as correct.

1. What is a linear combination?

A linear combination is a mathematical operation that involves adding together two or more quantities that have been multiplied by different coefficients. It is commonly used in statistics and data analysis to combine different variables or data points in order to create a new variable or data set.

2. How is a linear combination used in handling missing data?

Linear combination can be used to handle missing data by creating a new variable or data set that incorporates information from the existing variables. This can help to fill in the gaps caused by missing data and provide a more complete picture for analysis.

3. Can linear combination be used with categorical data?

Yes, linear combination can be used with categorical data. In this case, the coefficients used in the combination will represent the relative importance or weight of each category in the overall analysis.

4. How do you determine the coefficients for a linear combination?

The coefficients for a linear combination can be determined through various methods, such as statistical techniques like regression analysis or through expert knowledge. The coefficients should be chosen in a way that best represents the relationship between the variables being combined.

5. What are the advantages of using linear combination for missing data?

Linear combination can help to reduce the impact of missing data on statistical analysis by incorporating information from existing variables. It can also help to improve the accuracy and completeness of the data set, leading to more reliable results and conclusions.

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