# Linear algebra

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

1. 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.

posted by MathMate

First Name

## Similar Questions

1. ### 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? Why? If v1,...,v4 are in R^4 and v3 is not a linear combination of v1, v2, v4 then {v1,
2. ### Linear Algebra

Hello, could anyone help me with this excersise of linear algebra, Please? Well it's about linear transformation defined as T:V--W , the excersices give me a set of linearly independent vectors of V {v1,v2, v3... vk} and then asks
3. ### 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.
4. ### 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
5. ### 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 ,
6. ### 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
7. ### 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
8. ### 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? So I know how to find out if this set is linearly indep. But, I don't know how to use row rank to find
9. ### 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)
10. ### college Algebra/Linear Algebra

Find a Basis for each of these substances of R^4 (a) All vectors whose components are equal (b) All vectors whose component add to zero (c) All vectors that are perpendicular to (1,1,0,0) and (1,0,11) (d) The column space (In R^2)

More Similar Questions