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timon
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1. The problem statement
let \vec x and \vec y be linearly independent vectors in R^n and let S=\text{span}(\vect x, \vect y). Define the matrix A as
Show that N(A)=S^{\bot}.
2.equations
I have a theorem that saysN(A) = R(A^T)^{\bot}.
A is symmetric; A = A^T.
3.Plan of attack
From the given above, it follows that if i can proof that S is the orthogonal complement of R(A), i'll be done. To do that, i'll have to show that all elements of S are orthogonal to R(A), and that any vector orthogonal to R(A) is part of S.
Thus i want to show that
(I)the vectors \vec x and \vec y are orthogonal to any vector \vec z \in R(A)
(II)any vector \vec k that is orthogonal to all vectors \vec z \in R(A) can be written as a linear combination of \vec x and \vec y.
I'm really not seeing how to do this. Hope someone can help me out, or at least tell me if I'm on the right track. Cheers.
let \vec x and \vec y be linearly independent vectors in R^n and let S=\text{span}(\vect x, \vect y). Define the matrix A as
A=\vec x \vec y^T + \vec y \vec x^T.
Show that N(A)=S^{\bot}.
2.equations
I have a theorem that saysN(A) = R(A^T)^{\bot}.
A is symmetric; A = A^T.
3.Plan of attack
From the given above, it follows that if i can proof that S is the orthogonal complement of R(A), i'll be done. To do that, i'll have to show that all elements of S are orthogonal to R(A), and that any vector orthogonal to R(A) is part of S.
Thus i want to show that
(I)the vectors \vec x and \vec y are orthogonal to any vector \vec z \in R(A)
(II)any vector \vec k that is orthogonal to all vectors \vec z \in R(A) can be written as a linear combination of \vec x and \vec y.
I'm really not seeing how to do this. Hope someone can help me out, or at least tell me if I'm on the right track. Cheers.
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