Re: Determining Linear dependent vectors

• To: mathgroup at smc.vnet.net
• Subject: [mg66272] Re: [mg66248] Determining Linear dependent vectors
• From: "David Park" <djmp at earthlink.net>
• Date: Sat, 6 May 2006 01:55:07 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Here is a set of vectors in the rows.

vectors = {{1, 0, 3, 2, 0}, {0, 1, 2, 0, 3}, {1, 1, 5, 2, 3}, {2, -1, 4,
4, -3}, {1, 2, 7, 2, 6}};

Here is the null space of the transpose of vectors.

nmat = NullSpace[Transpose@vectors]
{{-1, -2, 0, 0, 1}, {-2, 1, 0, 1, 0}, {-1, -1, 1, 0, 0}}

Multiplying the null space vectors times vectors gives all zeros.

nmat.vectors
{{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}

We have three dependent vectors and we can solve for them in terms of the
first two independent vectors. Label the vectors {v1,v2,v3,v4,v5}. Then

Thread[nmat.{v1, v2, v3, v4, v5} == {0, 0, 0}]
Solve[%, {v3, v4, v5}]
{-v1 - 2 v2 + v5 == 0, -2 v1 + v2 + v4 == 0, -v1 - v2 + v3 == 0}
{{v5 -> v1 + 2 v2, v4 -> 2 v1 - v2, v3 -> v1 + v2}}

David Park

From: Saurabh [mailto:saurabh911 at gmail.com]
To: mathgroup at smc.vnet.net

Am looking for methods to determine linearly dependent vectors out of a
given set. Any pointers appreciated.

Thanks,

```

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