Check Linear Dependence/Independence of Vectors Without Calculator

[aliaze1]

Does anyone know a good way to check if a given set of vectors (assume we just know we have a set, not their values) is linearly dependent or linearly independent without a calculator?

Ex: Given a set of n-dimensional vectors, say, vector₁, vector₂, and vector₃, how would one determine if these vectors are linearly independent or dependent?

If I were to take the transpose of these vectors and make them into a matrix, and find the rank of this matrix, then I could perhaps check? If rank = number of vectors that make that matrix, then I have linear independence. If rank = less than number of vectors that make the matrix, then I have dependence (please correct me if I am wrong). I am a bit confused, as I know that the dimensions given (n) matter. If number of vectors is same as n/not same as n, does that make a difference?

Is there a better way of doing this? Graphically? Way to do this without looking at rank?

 

[micromass]

Does anyone know a good way to check if a given set of vectors (assume we just know we have a set, not their values) is linearly dependent or linearly independent without a calculator?

Ex: Given a set of n-dimensional vectors, say, vector₁, vector₂, and vector₃, how would one determine if these vectors are linearly independent or dependent?

If I were to take the transpose of these vectors and make them into a matrix, and find the rank of this matrix, then I could perhaps check? If rank = number of vectors that make that matrix, then I have linear independence. If rank = less than number of vectors that make the matrix, then I have dependence (please correct me if I am wrong).

This is correct. Just put the vectors in a matrix and calculate the rank. The rank is the maximal number of linear independent vectors. Thus if the rank = number of vectors, then you have independence, otherwise, you have dependence.

Also, there is no need to take the transpose of the matrix. The rank of a matrix equals the rank of the transpose.

I am a bit confused, as I know that the dimensions given (n) matter. If number of vectors is same as n/not same as n, does that make a difference?

No, the above procedure will work for any number of vectors. Note that the rank is defined for any matrix, not just square matrices!

But of course, if you have more vectors than your dimension, you will always have dependence! The rank of that matrix will also give you this.

Is there a better way of doing this? Graphically? Way to do this without looking at rank?

I think this is the best way of doing this. You can always draw the vectors and see whether they are dependent, but that becomes rather inconvenient if your dimension exceeds 3...