Science:Math Exam Resources/Courses/MATH307/December 2012/Question 03 (a)

MATH307 December 2012

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[hide] Question 03 (a)
Let $S=\left\{[x_{1},x_{2},x_{3}]^{T}:\,x_{1}+x_{2}+x_{3}=0\right\}$ be the subspace of vectors in $\mathbb {R} ^{3}$ whose components sum to zero. (a) Find a matrix A so that S is the null space of A, i.e., S = N(A).

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[show] Hint 1
How are the row space and the nullspace of of a matrix related?

[show] Hint 2
The nullspace of a matrix A is the orthogonal complement of the row space. Can you find a vector orthogonal to S?

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[show] Solution
The definition of S contains a dot product which reveals the vector orthogonal to S: ${\begin{aligned}S&=\left\{[x_{1},x_{2},x_{3}]^{T}:\,x_{1}+x_{2}+x_{3}=0\right\}\\&=\left\{[x_{1},x_{2},x_{3}]^{T}:\,{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}}\cdot {\begin{bmatrix}1\\1\\1\end{bmatrix}}=0\right\}\end{aligned}}$ Since the vector ${\begin{bmatrix}1\\1\\1\end{bmatrix}}$ is orthogonal to S, and the nullspace of A ( which is S) is orthogonal to the row space of A, we can choose any matrix A whose row space is spanned by ${\begin{bmatrix}1\\1\\1\end{bmatrix}}$ . Such as $A={\begin{bmatrix}1&1&1\end{bmatrix}}$ or $A={\begin{bmatrix}1&1&1\\2&2&2\\0&0&0\\\pi &\pi &\pi \end{bmatrix}}.$