Prove that vj must be the zero vector

**wopashui** · April 6th 2010, 11:55 AM

Let S = { $v_1, v_2,....v_k$ } be an orthogonal set of vectors in $R^n$ . If S is linearly dependent, prove that one of the $v_j$ must be the zero vector.

Find an orthonormal basis for the column space of the matrix A =
[2 5 7]
[3 1 8]
[6 6 10]
[0 6 -9] and obtain the QR factorisation of A.

**nimon** · April 7th 2010, 01:42 AM

Let $n=3$ , and let $v_{1}=(1,0,0),\,\, v_{2}=(0,1,0)$ . Then $\{v_{1},v_{2}\}\subset \mathbb{R}^{n}$ are orthogonal and neither are zero. Did you mean to assume that $k>n$ ?

**HallsofIvy** · April 7th 2010, 03:42 AM

That set is not linearly dependent.

wopashui, if the set is linearly dependent, then there exist numbers, $a_i$ , not all 0, such that $a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_kv_k= 0$ . Now take the dot product of both sides of that with $v_1$ , $v_2$ , etc.

As for the second problem, think of the three columns of A as three vectors and use "Gram-Schmidt" to find an orthonormal basis.

**nimon** · April 7th 2010, 07:08 AM

Woops! Note to self: read the question!

**HallsofIvy** · April 7th 2010, 12:57 PM

Originally Posted by nimon

Woops! Note to self: read the question!

Always a good suggestion!

**wopashui** · April 13th 2010, 11:15 AM

Originally Posted by HallsofIvy

That set is not linearly dependent.

wopashui, if the set is linearly dependent, then there exist numbers, $a_i$ , not all 0, such that $a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_kv_k= 0$ . Now take the dot product of both sides of that with $v_1$ , $v_2$ , etc.

As for the second problem, think of the three columns of A as three vectors and use "Gram-Schmidt" to find an orthonormal basis.

but none of the vector of A is orthonormal, whhich vector do I start with? And what is the QR factorisation of A?

**nimon** · April 13th 2010, 11:48 AM

It doesn't matter that the columns aren't orthogonal. The Gram-Schmidt procedure takes any set of basis vectors and turns them into an orthogonal basis of the same set, which can then be normalised to give an orthonormal set.

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