Linear Independence of vectors question

In summary, the problem involves finding the coefficients of the vectors A, B, and C, given that they are not linearly independent. By taking the dot product of the equation a_1A + a_2B + a_3C = 0 with each of the vectors, a system of equations can be obtained. By arbitrarily choosing a value for one of the coefficients, the remaining two can be solved for using substitution.
  • #1
bossman007
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0

Homework Statement



Suppose that A, B and C are not linearly independent. Then show how the [itex]a_i[/itex] can be computed, up to a common factor, from the scalar products of these vectors with each other

Homework Equations



[itex]a_1[/itex]A + [itex]a_2[/itex]B + [itex]a_3[/itex]C = 0

[itex]a_1[/itex]=[itex]a_2[/itex]=[itex]a_3[/itex]=0

Hint - Suppose that there are non-zero values of the [itex]a_i[/itex]'s that satisfy
[itex]a_1[/itex]A + [itex]a_2[/itex]B + [itex]a_3[/itex]C = 0. Then, taking the dot product of both sides of this equation with A will yield a set of equations that can be solved for the [itex]a_i[/itex]'s

The Attempt at a Solution



[itex]a_1[/itex]AA + [itex]a_2[/itex]BA + [itex]a_3[/itex]CA=0

no idea where to go from here, I took the dot product of both sides but confused from the wording of the question what my next step should be, or If I did my dot product right
 
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  • #2
Do the same with B and C. The scalar product are numbers, so you have a system of equations for the three unknown parameters. As the vectors are not independent, one of the parameter can be chosen arbitrary. Solve the system of equations for the other two coefficients.

ehild
 
  • #3
Thank you for your reply. you're saying arbitrarily choose a value? If so, I chose [itex]a_1[/itex]=1

from that, my set of equations looks like, after moving the AA , AB and AC to the other side of the equation i get this:

[itex]a_2[/itex]AB + [itex]a_3[/itex]AC = -AA
[itex]a_2[/itex]BB + [itex]a_3[/itex]BC = -AB
[itex]a_2[/itex]BC + [itex]a_3[/itex]CC = -AC

**any vector combination above is a dot product, I just didnt know how to latex code it***

I went on to try substitution to solve for [itex]a_2[/itex], but the result was messy and didnt seem like I was on the right track. Am I on the right track?
 
  • #4
It will not be that messy. Multiply the first equation with BC, the second equation with AC. Subtract them. a3 cancels and you can isolate a2.

ehild
 

FAQ: Linear Independence of vectors question

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

Linear independence of vectors refers to the property of a set of vectors where none of the vectors can be expressed as a linear combination of the others. In other words, no vector in the set can be written as a linear combination of the remaining vectors.

2. How can we determine if a set of vectors is linearly independent?

To determine if a set of vectors is linearly independent, we can use the determinant test or the rank-nullity theorem. The determinant test involves creating a matrix with the vectors as columns and taking the determinant. If the determinant is non-zero, the vectors are linearly independent. The rank-nullity theorem states that if the rank of the matrix formed by the vectors is equal to the number of vectors, then the vectors are linearly independent.

3. What is the importance of linear independence of vectors in linear algebra?

Linear independence is an important concept in linear algebra because it allows us to determine if a set of vectors can be used as a basis for a vector space. Linearly independent vectors form a basis for a vector space, which means they can span the entire space and any vector in the space can be expressed as a linear combination of these vectors.

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

Yes, a set of two vectors can be linearly independent. For example, the vectors (1,0) and (0,1) in two-dimensional space are linearly independent because neither can be written as a multiple of the other.

5. What is the difference between linear independence and orthogonality?

Linear independence and orthogonality are two different concepts in linear algebra. Linear independence refers to the relationship between vectors and whether they can be expressed as linear combinations of each other. Orthogonality, on the other hand, refers to the relationship between vectors where they are perpendicular to each other. Two vectors can be linearly independent and not orthogonal, or they can be both linearly independent and orthogonal.

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