Linear algebra

posted by .

Find two vectors v and w such that the three vectors u = (1,-1,-1), v and w are linearly independent independent.

  • Linear algebra -

    Three vectors are linearly independent if the determinant formed by the vectors (in columns) is non-zero.
    So for u=(1,-1,-1), v=(a,b,c), w=(d,e,f)
    There are many possible choices of v and w such that the determinant
    1 a d
    -1 b e
    -1 c f
    is non-zero.

    The simplest way is to create a triangular matrix such that the diagonal is all non-zero, or
    1 0 0
    -1 b 0
    -1 c f

    where b and f are non-zero, and the determinant evaluates to b*f≠0.

    Example: (b=-1,c=1,f=1)
    (1,-1,-1),(0,-1,1),(0,0,1) are linearly independent because the determinant
    1 0 0
    -1 -1 0
    -1 1 1
    evaluates to 1*(-1)*(-1)=-1 ≠ 0

    Note:
    vectors that are orthogonal to each other are linearly independent, since each cannot be a linear combination of the others.
    However, linear independent vectors need not be orthogonal. Therefore their dot products need not be zero.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. math

    If A^TA is an invertible matrix, prove that the column vectors of A are linearly independent. You know that if statement X implies statement Y then that is equivalent to Not(Y) implies Not(X). You can start by taking the column vectors …
  2. Maths-Vectors Help!

    Please can you help me as I have just been introduced to your Help Forum: Determine whether the vectors u, v and w given below are linearly independent or dependent where u, v and w are non-colliner vectors such that u=2a-3b+c , v=3a-5b+2c …
  3. algebra

    If v1,...,v4 are in R^4 and v3 is not a linear combination of v1, v2, v4 then {v1, v2, v3, v4] is linearly independent. Is this true or false?
  4. linear algebra

    which of the following sets of vectors span R^3?
  5. math

    Find an orthonormal basis for the subspace of R^3 consisting of all vectors(a, b, c) such that a+b+c = 0. The subspace is two-dimensional, so you can solve the problem by finding one vector that satisfies the equation and then by constructing …
  6. math

    Find the least squares approximation of x over the interval [0,1] by a polynomial of the form a + b*e^x --------------------------------------------------------- The polynomial produces an output space with two linearly independent …
  7. Algebra

    If W1,w2,w3 are independent vectors, show that the sums V1=W2+W3, V2=W1+W3 and V3=W1+W2 are independent . (Write C1V1+C2V2+C3V3=0 in terms of the W¡¦s. Find and solve equation for the C¡¦s)
  8. Linear Algebra

    Prove that If a vector space is of dimension n and a set of vectors spans V, then that set of vectors must be linearly independent.
  9. linear algebra

    Solve using the concept of rank. Is S={−16 −7 −21,2 1 3, 21 9 2} a linearly independent set of vectors in R3?
  10. Linear Algebra

    Hello, could anyone help me with this excersise of linear algebra, Please?

More Similar Questions