Let(adsbygoogle = window.adsbygoogle || []).push({}); u= [1, 2, 3, -1, 2]^{T},v= [2, 4, 7, 2, -1]^{T}inℝ^{5}.

Find a basis of a spaceWsuch thatw⊥uandw⊥vfor allw∈W.

I think the question is quite easy. Given this vectorwin the space W is orthogonal to bothuandv. I can only think ofwbeing a zero vector. But would this be too trivial?

=w^{T}u= 0w^{T}v

w(^{T}u-v) = 0

I believe there is something wrong. Things above are what I can compute...

Is there any other way to solve this question?

-----------------------------------------------------------------------------------------------------

Oh, I've thought of another way.

Set a augmented matrix

[1 2 3 -1 2 | 0]

[2 4 7 2 -1 | 0]

~

[1 2 3 -1 2 | 0]

[0 2 4 3 -3 | 0]

Let x_{5}= t, x_{4}= s, x_{3}= z where t, s, z ∈ ℝ.

∴ x_{2}= 1.5t - 1.5s - 2z

x_{1}= -5t + 4s + z

Therefore, the solution set is {t[-5 1.5 2 0 0 1]^{T}+ s[4 -1.5 0 1 0]^{T}+ z[1 -2 1 0 0]^{T}}.

The basis is {[-5 1.5 2 0 0 1]^{T}, [4 -1.5 0 1 0]^{T}, [1 -2 1 0 0]^{T}}.

Is this correct?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Finding a basis of a space

Tags:

Loading...

Similar Threads for Finding basis space |
---|

I How to find admissible functions for a domain? |

I Diagonalization and change of basis |

I Measures of Linear Independence? |

I How to find the matrix of the derivative endomorphism? |

I Finding the Kernel of a Matrix Map |

**Physics Forums | Science Articles, Homework Help, Discussion**