What is the fastest way of calculating whether vectors are linearly independent?Is the following family ( u 1 , u 2 , u 3 ) linearly independent? u 1 := ( 1 3 5 − 1 ) , u 2 := ( 1 − 1 − 3 3 ) , u 3 := ( 3 2 1 4 ) So, do I have to form a matrix U with columns u 1 , u 2 and u 3 and use gaussian elimination to see whether in every column is a pivot or is there a faster way?

Question

What is the fastest way of calculating whether vectors are linearly independent?Is the following family   (      u    1    ,      u    2    ,      u    3    ) linearly independent?      u    1    :=      (                            1                                      3                                      5                                      −          1                      )    ,      u    2    :=      (                            1                                      −          1                                      −          3                                      3                      )    ,      u    3    :=      (                            3                                      2                                      1                                      4                      )  So, do I have to form a matrix U with columns       u    1  ,       u    2   and       u    3   and use gaussian elimination to see whether in every column is a pivot or is there a faster way?

Prezrenjes0n · Accepted Answer

I&#039;m not sure this is any faster, but here&#039;s another approach.Clearly, no two of these vectors are linearly dependent. Thus, if these three vectors are linearly dependent, then       u    3   must be a linear combination of       u    1   and       u    2  . Suppose       u    3    =  α      u    1    +  β      u    2  . Looking at the first entry, we get   α  +  β  =  3. Looking at the second entry, we get   3  α  −  β  =  2. Together, these two equations imply that   α  =  5      /    4 and   β  =  7      /    4. Using these values for   α and   β, we confirm that   5  α  −  3  β  =  1 and   −  α  +  3  β  =  4. Therefore,      u    3    =      5    4        u    1    +      7    4        u    2  so these three vectors are linearly dependent.

piopiopioirp · Accepted Answer

With Gaussian elimination, it is not very long to prove the rank is 2:                              [                                                    1                                            1                                            3                                                                    3                                            −                1                                            2                                                                    5                                            −                3                                            1                                                                    −                1                                            3                                            4                                              ]                ⇝                  [                                                    1                                            3                                            1                                                                    −                1                                            2                                            3                                                                    −                3                                            1                                            5                                                                    3                                            4                                            −                1                                              ]                ⇝                  [                                                    1                                            3                                            1                                                                    0                                            5                                            4                                                                    −                3                                            1                                            5                                                                    0                                            5                                            4                                              ]                ⇝                  [                                                    1                                            3                                            1                                                                    0                                            5                                            4                                                                    0                                            10                                            8                                                                    0                                            0                                            0                                              ]                ⇝                  [                                                    1                                            3                                            1                                                                    0                                            5                                            4                                                                    0                                            0                                            0                                                                    0                                            0                                            0                                              ]

Is the following family (u_1,u_2,u_3) linearly independent? u_1 :=((1),(3),(5),(-1)), u_2 :=((1),(-1),(-3),(3)), u_3 :=((3),(2),(1),(4))

Answered question

Answer & Explanation

New Questions in Precalculus