$Linear Algebra Help : Consider Two Planes, P1 and P2$

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$Linear Algebra Help : Consider Two Planes, P1 and P2$

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Linear Algebra

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(a) Let $n_1$ and $n_2$ be the normal vectors of the two planes $P_1$ and $P_2$, respectively. Define the vector
\[u= n_1 \times n_2.\]
Recall that the cross product $n_1\times n_2$ is orthogonal to the normal vectors $n_1$ and $n_2$, thus the vector $u$ lies on both planes $P_1$ and $P_2$.

(b) Since the matrices $A$ and $B$ are of rank 1, the null space of $L_1(x)=Ax$ and $L_2(x)=Bx$ ($L_1,L_2: \R^3 \rightarrow \R^3$ are linear) are two dimensional vector spaces. Indeed if we define
\[P_1=\{x:\in R^3: L_1(x)=Ax=0\} \text{and} P_2=\{x:\in R^3: L_2(x)=Bx=0\} \]
Then $P_1, P_2$ are two dimesnional subspace of $R^3$, which are indeed planes. From part (a) we know that there is a $v$ that lies on both $P_1$ and $P_2$, which means
\[L_1(v)=Av=0 \text{and} L_2(v)=Bv=0. \]

Erdos

Erdos

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Let me know if you need any clarifications.
Matchmaticians Alumnus

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Aren't planes two dimensional?
Erdos

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Yes, you are right. I had the co-dimension in mid. I have revised the solution accordingly.

$Linear Algebra Help : Consider Two Planes, P1 and P2$

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