Linear Independence: det[v, u, w] = 0 iff k ≠ ___?

In summary, the vectors v, u, and w are linearly independent if and only if k does not equal 15. The determinant of v, u, and w must be set to 0 and solved for k, resulting in a simple linear equation to determine the value of k.
  • #1
snoggerT
186
0
the vectors: v= [-5, -8, 7], u= [2, 4, (-17+k)] and w= [2, 7, 1]
are linearly independent if and only if k does not equal ___?

- note that the vectors are supposed to be setup vertically with only one column and 3 rows.



det[v, u, w]



The Attempt at a Solution



- I tried setting up the determinant = 0 and then solving for k, but that doesn't seem to give me the right answer. I'm really not sure where to go with this type of problem. Please help.
 
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  • #2
Your method is correct. You should get a simple linear equation to solve for k. What did you get?
 
  • #3
Nevermind. I had my sign wrong. It was 15. Thanks for the help.
 
Last edited:

FAQ: Linear Independence: det[v, u, w] = 0 iff k ≠ ___?

1. What is linear independence?

Linear independence refers to a set of vectors in a vector space that cannot be written as a linear combination of the other vectors in the set. In other words, none of the vectors in the set can be expressed as a linear combination of the others.

2. How is linear independence determined?

Linear independence is determined by calculating the determinant of the vectors in a set. If the determinant is equal to zero, then the vectors are linearly dependent. If the determinant is not equal to zero, then the vectors are linearly independent.

3. What does det[v, u, w] = 0 signify?

When the determinant of a set of vectors is equal to zero, it signifies that the vectors are linearly dependent. This means that at least one of the vectors in the set can be expressed as a linear combination of the others.

4. Is det[v, u, w] = 0 the only way to determine linear independence?

No, there are other methods for determining linear independence such as checking for zero coefficients in a linear combination, or using the rank of a matrix to determine if the vectors are linearly independent.

5. What is the significance of k ≠ ___ in the statement det[v, u, w] = 0 iff k ≠ ___?

The value of k represents a scalar that is used in a linear combination of the vectors. The statement det[v, u, w] = 0 iff k ≠ ___ means that for a set of vectors to be linearly independent, the determinant must be equal to zero for all values of k except for one specific value, which is represented by the blank space. This specific value is known as the critical value of k.

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