how to determine linearly dependence of vectors? (1 Viewer)

[bubb]

How to determine if vectors a= -5j + k , b= 3i+4j-5k and c= 2i +j-3k are linearly dependent??

I tried addition of the three vectors = 0 but it didn't work?

 

[seanieg89]

Three column vectors are linearly dependent if and only if the matrix with these three columns has vanishing determinant.

 

[braintic]

How to determine if vectors a= -5j + k , b= 3i+4j-5k and c= 2i +j-3k are linearly dependent??

I tried addition of the three vectors = 0 but it didn't work?

Adding the three vectors is not enough. You have to guarantee any linear combination of the vectors will never be zero for them to be linearly independent.
ie. can you find c1, c2, c3 for which c1(0, -5, 1) + c2(3, 4, -5) + c3(2, 1, -3) is zero?

 

[FrankXie]

I tried addition of the three vectors = 0 but it didn't work?

not the simple addition, should be the linear combination, how about try 2b-3c+a ?

 

[Drongoski]

The 3 vectors are linearly dependent.

e.g. 1(-5j + k) + 2(3i + 4j - 5k) -3(2i + j - 3k) = 0

or:-5( -5j + k) - 10(3i + 4j - 5k) + 15(2i + j - 3k) = 0

So: it means you can generate any vector from the other 2 by taking a suitable linear combination of these 2 vectors; i.e. a multiple of one + a multiple of the second, where these multiples are constants.

 

[seanieg89]

The 3 vectors are linearly independent.

e.g. 1(-5j + k) + 2(3i + 4j - 5k) -3(2i + j - 3k) = 0

or:-5( -5j + k) - 10(3i + 4j - 5k) + 15(2i + j - 3k) = 0

So: it means you can generate any vector from the other 2 by taking a suitable linear combination of these 2 vectors; i.e. a multiple of one + a multiple of the second, where these multiples are constants.

Typo.

 

[Drongoski]

Typo.

Thanks Sean; corrected.