Friday

March 27, 2015

March 27, 2015

Posted by **Neeraj** on Thursday, August 11, 2011 at 7:59am.

- Linear algebra -
**MathMate**, Thursday, August 11, 2011 at 9:01amThree 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.

**Answer this Question**

**Related Questions**

alegbra - If v1,...,v4 are in R^4 and v3 is not a linear combination of v1, v2, ...

Linear Algebra - Prove that If a vector space is of dimension n and a set of ...

math - Find the least squares approximation of x over the interval [0,1] by a ...

Maths-Vectors Help! - Please can you help me as I have just been introduced to ...

math - If A^TA is an invertible matrix, prove that the column vectors of A are ...

math - Find an orthonormal basis for the subspace of R^3 consisting of all ...

Algebra - If W1,w2,w3 are independent vectors, show that the sums V1=W2+W3, V2=...

linear algebra - which of the following sets of vectors span R^3? a.){(1, -1, 2...

Calculus - Name two sets of vectors that could be used to span a set in R^3. ...

college Algebra/Linear Algebra - Find a Basis for each of these substances of R^...