2. Find a vector $x$ orthogonal to the row space of $A$, and a vector $y$ orthogonal to the column space, and a vector $z$ orthogonal to the nullspace: $A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 4 & 3 \\ 3 & 6 & 4 \end{bmatrix}$
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The row space of A is spanned by the rows of A. Row1 = [1 2 1] Row2 = [2 4 3] Row3 = [3 6 4] We can see that Row2 = 2*Row1 + 1 and Row3 = 3*Row1 + 1. Therefore, the row space of A is spanned by [1 2 1]. Show more…
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Find a vector $x$ orthogonal to the row space of $A$, and a vector $y$ orthogonat to the column space, and a vector $z$ orthogonal to the nullspace: $$ A=\left[\begin{array}{lll} 1 & 2 & 1 \\ 2 & 4 & 3 \\ 3 & 6 & 4 \end{array}\right] $$
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Find a vector x orthogonal to the row space of A, and a vector y orthogonal to the column space, and a vector orthogonal to the nullspace:
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(a) With the preceding $A$, use elimination to solve $A x=0$. (b) Show that the nullspace you just computed is orthogonal to $\boldsymbol{C}\left(A^{\mathrm{H}}\right)$ and not to the usual row space $C\left(A^{\mathrm{T}}\right)$. The four fundamental spaces in the complex case are $N(A)$ and $C(A)$ as before, and then $N\left(A^{\text {H }}\right)$ and $C\left(A^{\text {H }}\right)$.
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