Question

Let $A$ be a $2\times 3$ matrix of rank 2 with rows ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$. Show that $$\begin{gathered} P= \\ XA\mid X=[xy];x,y\text{ arbitrary } \end{gathered}$$ is the plane through the origin with normal ${\mathbf{r}}_1\times {\mathbf{r}}_2$.

Short answer

$P$ is a plane through the origin with normal ${\mathbf{r}}_1\times {\mathbf{r}}_2$.

Step-by-step solution

Understanding the Matrix Multiplication

To verify that $P=\{XA\mid X=[xy];x,y\text{ arbitrary}\}$ is a plane, observe that for a matrix $X$ with any arbitrary real numbers $x$ and $y$, $XA$ results in a linear combination of the rows ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ of matrix $A$.

Expressing the Condition for the Plane

The expression $XA$ implies that any point $\mathbf{p}\in P$ takes the form $x{\mathbf{r}}_1+y{\mathbf{r}}_2$. This tells us that $P$ contains all linear combinations of the vectors ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$, forming a plane through the origin since the rank of $A$ is 2.

Determining the Normal Vector

To find the normal vector to the plane formed by ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$, compute their cross product: ${\mathbf{r}}_1\times {\mathbf{r}}_2$. This vector is perpendicular to both ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$, thus it serves as the normal vector to the plane.

Confirming the Plane's Properties

Since the normal vector $\mathbf{n}={\mathbf{r}}_1\times {\mathbf{r}}_2$ is perpendicular to any vector in the plane $P$, every vector $x{\mathbf{r}}_1+y{\mathbf{r}}_2$ in the plane satisfies: $\mathbf{n}\cdot (x{\mathbf{r}}_1+y{\mathbf{r}}_2)=0$. This confirms that $P$ is indeed the plane through the origin with normal ${\mathbf{r}}_1\times {\mathbf{r}}_2$.

Key concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that involves multiplying matrices to produce a new matrix. Specifically, when you multiply a matrix by a vector or another matrix, you're essentially performing a series of dot products between the rows of the first matrix and the columns of the second.
For a matrix multiplication to be valid, the number of columns in the first matrix must be equal to the number of rows in the second matrix. For example, if you have a matrix $A$ with size $2\times 3$ (2 rows and 3 columns), it can be multiplied by another matrix or vector where the number of rows is 3.

• More formally, if $A$ is a $m\times n$ matrix, and $B$ is an $n\times p$ matrix, then the resulting matrix $AB$ will have dimensions $m\times p$.
• The element in the $i$-th row and $j$-th column of $AB$ is calculated by summing the products of the corresponding elements from the $i$-th row of $A$ and $j$-th column of $B$.

Matrix multiplication enables you to express transformations, solve systems of equations, and describe linear combinations, which we'll explore next.

Linear Combination

A linear combination is an expression constructed from a set of elements (e.g., vectors) by multiplying each element by a constant and adding the results. In linear algebra, linear combinations are used to express more complex vectors as combinations of simpler ones.
For instance, if you have vectors ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n$, a linear combination of these vectors can be written as $c_1{\mathbf{v}}_1+c_2{\mathbf{v}}_2+\dots +c_n{\mathbf{v}}_n$, where $c_1,c_2,\dots ,c_n$ are scalars.

• In the context of the exercise, any element $XA$ from the set $P$ can be expressed as a linear combination of the rows ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ of the matrix $A$.
• This means $x{\mathbf{r}}_1+y{\mathbf{r}}_2$ forms a plane, as it involves all possible linear combinations when $x$ and $y$ are varied freely.

This forms a foundational concept as it helps establish the formation of a plane through the origin by assembling combinations of basis vectors.

Cross Product

The cross product is a binary operation on two vectors in three-dimensional space, producing a third vector that is perpendicular to the two input vectors. The cross product of vectors $\mathbf{a}$ and $\mathbf{b}$ is denoted as $\mathbf{a}\times \mathbf{b}$.
The resulting vector from a cross product not only has a direction but also a magnitude that reflects the area of the parallelogram formed by the original vectors. The direction is determined by the right-hand rule.

• Mathematically, for vectors $\mathbf{a}=[a_1,a_2,a_3]$ and $\mathbf{b}=[b_1,b_2,b_3]$, the cross product $\mathbf{a}\times \mathbf{b}=[a_2{b}_3-a_3{b}_2,a_3{b}_1-a_1{b}_3,a_1{b}_2-a_2{b}_1]$.
• In this exercise, the cross product ${\mathbf{r}}_1\times {\mathbf{r}}_2$ results in a normal vector, perpendicular to the plane formed by vectors ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$.

This is particularly useful for defining planes in 3D space, ensuring the perpendicularity required for identifying normal vectors.

Normal Vector

In geometry and vector algebra, a normal vector to a plane or surface is one that is perpendicular to that surface. Identifying a normal vector is essential when describing planes, as it provides information about the plane's orientation in space.
For a plane defined by two vectors, its normal vector can be found using the cross product of those vectors. In the context of the original exercise, for the plane described by the vectors ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$, the normal vector is ${\mathbf{r}}_1\times {\mathbf{r}}_2$.

• This vector is crucial because any vector lying on the plane should be orthogonal to the normal vector.
• In simple terms, the normal vector ensures that the dot product with any vector within the plane equals zero: $\mathbf{n}\cdot (x{\mathbf{r}}_1+y{\mathbf{r}}_2)=0$.

The role of the normal vector is pivotal for calculations related to projections, reflections, and understanding the spatial geometry of surfaces.