Q01 are linearly independent vectors, so are....

In summary, the conversation discusses the definition of "linearly independent" and attempts to prove that if $v_1, v_2, v_3$ are linearly independent, then so are $Av_1, Av_2, Av_3$. The question also raises the importance of knowing what "A" represents, as this proof is not necessarily true for a general linear transformation but only for an invertible one.
  • #1
karush
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MHB
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Let A be invertible. Show that, if $\textbf{$v_i,v_2,v_j$}$ are linearly independent vectors, so are \textbf{$Av_1,Av_2,Av_3$}

https://drive.google.com/file/d/1OuHxfUdACbpK4E5aca2oBzdaxGR0IYKv/view?usp=sharing
p57.png


ok I think this is the the definition we need for this practice exam question,
However I tried to insert using a link but not successful
I thot if we use a link the image would always be there unless we delete its source

as to the question... not real sure of the answer since one $c_n$ may equal 0 and another may not

Anyway Mahalo...
 
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  • #2
Yes, that is the definition of "linearly independent" you need. Now, what are you trying to prove?

You have "Show that, if $v_1$, $v_2$, $v_3$ are linearly independent, then so are $Av_1$, $Av_2$, $Av_3$" but what is "A"? If it is a general linear transformation this is not true. If A is an INVERTIBLE linear transformation then it is true and can be shown by applying $A^{-1}$ to both sides of$x_1Av_1+ x_2Av_2+ x_3Av_3= 0$.
 

1. What does it mean for vectors to be linearly independent?

Linear independence refers to a set of vectors that cannot be written as a linear combination of each other. In other words, none of the vectors can be expressed as a combination of the others, making them unique and essential to the set.

2. How can I determine if a set of vectors are linearly independent?

To determine if a set of vectors are linearly independent, you can use the linear independence test. This involves setting up a system of equations using the vectors and solving for the coefficients. If the only solution is when all coefficients are equal to zero, then the vectors are linearly independent.

3. What is the significance of linearly independent vectors?

Linearly independent vectors are important in linear algebra and other areas of mathematics because they form a basis for vector spaces. This means that any vector in the space can be expressed as a unique combination of the linearly independent vectors.

4. Can a set of linearly independent vectors be linearly dependent?

No, by definition, linearly independent vectors cannot be linearly dependent. If a set of vectors is linearly dependent, it means that at least one of the vectors can be expressed as a linear combination of the others, making them not unique and therefore not linearly independent.

5. How are linearly independent vectors used in scientific research?

Linearly independent vectors have various applications in scientific research, particularly in fields such as physics, engineering, and computer science. They are used in data analysis, modeling, and solving systems of equations. In addition, they are essential in understanding and manipulating vector spaces in mathematics.

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