Eroxl's Notes
Web Work 3 (MATH 221)

Problem 1

Are the vectors , , and linearly independent? If they are linearly dependent, find scalars that are not all zero such that they sum to If they are linearly independent, find the only scalars that will make them sum to .

Problem 2

Determine which of the following sets of vectors are linearly independent and which are linearly dependent.

(a). And

They are linearly dependent as the first is a multiple of the second

(b). , , And

They are linearly dependent.

(c). , , , , And

The vectors are only and there are 5 of them so they're linearly dependent.

(d). And

The last rows are irreconcilable so they are linearly independent.

(e). , , ,

The 3rd vector is zero so they must be linearly dependent

(f). , , And

They are linearly independent

Problem 3

Which of the following sets of vectors are linearly independent?

  • [ ] A. { ( -5, -9 ), ( -6, 4 ) , ( -3, -8 ) }

The dimensionality of the vectors is less then the length so they're dependent

  • [x] B. { ( 3, 2, 4 ), ( 0, 1, -6 ), ( 7, 8, 5 ) }

  • [x] C. { ( -7, -1, -8 ), ( 1, -2, -3) }
  • [ ] D. { ( 6, 3, -5 ), ( -9, -6, 4 ), ( 3, 3, 1 ) }

The first vector plus the third is equal to the second so they are dependent.

  • [x] E. { ( 4, 3, -1, -8 ), ( 1, 1, -6, -9) }
  • [ ] F. { ( 6, -4 ), ( 0, 0 ) }
  • [ ] G. { ( 9, -4, 0 ), ( 5, 2, 0 ), ( 7, 8, 0 ) }
  • [x] H. { ( 1, 5 ), ( 9, 1 ) }

Problem 4

Let be a set of vectors. If determine whether or not is linearly independent. If is dependent enter a non-trivial linear relation or 0 if it is independent.

is linearly dependent and the coefficients are .

Problem 5

Suppose is a set of linearly independent vectors. If determine whether is a linearly independent set. If is dependent enter a non-trivial linear relation or 0 if it is independent.

It i's linearly independent as we know no combination of and can equal so thus no combination of those two can equal . Therefore the only relationship that holds is the trivial .

Problem 6

Let be a linearly independent set of vectors. Select the best statement.

  • [x] A. is always a linearly independent set of vectors.
  • [ ] B. could be a linearly independent or linearly dependent set of vectors depending on the vectors chosen.
  • [ ] C. is never a linearly independent set of vectors.
  • [ ] D. none of the above

Problem 7

Let , and be three linearly independent vectors in . Determine whether the following sets of vectors are linearly independent or dependent.

(a). The Set

They are dependent as any multiple of yields the zero vector.

(b). The Set

Reversing the signs from (a) we can clearly see that the relationship can only be zero if , and which is inconsistent with unless they are all zero.

Therefore the set is linearly independent.

Problem 8

Let , and be three linearly independent vectors in . Determine the value of , such that the set is linearly dependent.

Problem 9

Determine if the subset of consisting of vectors of the form , where is a subspace. Select true or false for each statement.

(a). This Set is Closed under Vector addition

Consider the vectors and their sum is not in the subspace, therefore it is not closed under vector addition.

(b). This Set is Closed under Scalar Multiplications

If then for any constant the following must be true . Therefore it is closed under scalar multiplication.

(c). This Set is a Subspace

The set is not closed under vector addition so it can not be a subspace.

(d). The Set Contains the Zero Vector

For , , and therefore the set contains the zero vector.