Vectors: Algebra and Geometry

Vectors

Vectors in \(R^n\)

We use \(R\) to represent real numbers.

\(R^n\)=all ordered \(n\)-tuples of real numbers \((x_1,x_2,\dots,x_n)\).
An \(n\)-tuple of reals is called \(R^n\).

In any \(R^n\), the point conresponding to the \(n\)-tuple \((0,0,\dots,0)\) is special. We call it the origin.

Vectors and geometry

When \(n=2\) or \(3\), we can represent these lists geometrically in two ways.

Example (Euclidean plane), when \(n=2\), we can think of \(R^2\) as the \(xy\)-plane.

Example (\(3\)-space), when \(n=3\), we can think of \(R^2\) as space we live in. Wa can do so because every point can be represented by an ordered triple of reals, namely, its \(x-\),\(y-\) and \(z-\)coordinates.

Vectors and Points. A vector of an element of \(R_n\), especially when drawn as an arrow.

The \(R^2\) vector from \((1,1)\) to \((2,3)\) is the same as the vector from \((0,0)\) to \((1,2)\), they are considered equally because they both represent the same arrow(vector) which is \((1,2)\).

Based vectors. When one vector \(v\) is considered as ans arrow from origin to a point \(p\). In this case, the coordinates of \(v\) and \(p\) are the same.

Even though we can't draw elements of \(R^4\) and \(R^5\), \(\dots\) geometrically, we still refer to them as points or vectors. We have to rely to algebra but not geometric intuition.

Such as we can use a \(R^4\) vector \((x,y,z,w)\) to represent the traffic flow of the point \((x,y,z)\) is \(w\), and a \(R^6\) vector \((x,y,z,r,g,b)\) to represent the color of a point on a cube, additionally, a QR-code is a matrix of \(29\times 29\), so we can indicate a QR-code as a \(R^{841}\), the number of each represents the color on each point is whether black or white.

Vectors Algebra and Geometry

We can represent a based vector as a coordinate of its end point. What is the operation calculate between vectors.

We can add two vectors together provided they have the same number of entries: \((x_1,y_1,z_1)+(x_2,y_2,z_2)=(x_1+x_2,y_1+y_2,z_1+z_2)\).

We can multiply or scale a vector with a real \(\lambda\): \(\lambda(x,y,z)=(\lambda x,\lambda y,\lambda z)\).

The geometrical meaning of the addition of two \(R^2\) vectors:

and samely subtraction:

and scalar multipilication:

you can easily discover that a scalar multipilication is a kind of stretching or shrotening of a vector, otherwise maybe change its direction to the opposite if \(\lambda\) is a negative.

Linear Combination

Linear combination is the operation of merge basic vectors to represent complicated vectors.

In a second demention, we can use \(\overrightarrow i=(1,0)\) and \(\overrightarrow j(0,1)\) to represent all \(R^2\) vectors as \(\alpha \overrightarrow i+\beta \overrightarrow j\), for example \((3,-5)=3\overrightarrow i+(-5)\overrightarrow j\).

So the basic of linear combination is to represent vectors of \(R^n\) \(\alpha=(\alpha_1,\alpha_2,\dots,\alpha_n)\) by scalar multipilication and addition of the \(n\) base vectors \((1,0,\dots,0)\), \((0,1,\dots,0)\), \(\dots\), \((0,\dots,0,1)\).

Vector Equations and Spans

Vector Equations

It is easy to determine whether a vector \(a\) is equal to \(b\).

We can use vectors scalar multiplication to indicate some equations. For example, the equation:
\(x(1,2,6)+y(-1,-2,-1)=(8,16,3)\)
simplifies to
\((x-y,2x-2y,6x-y)=(8,16,3)\).

For two vectors to be equal, all the coordinates must be equal, so this is just the system of linear equations

\[\begin{cases} x-y=8\\ 2x-2y=16\\ 6x-y=3 \end{cases} \]

But it may not have solution or maybe have mutiple solutions.

Spans

With some vectors of \(R^n\), namely \(v_1,v_2,\dots,v_k\), the spans of them is the set of all vectors that can be represented by \(v_1,v_2,\dots,v_k\), in symbols, \(\operatorname{Span}(v_1,v_2,\dots,v_k)=\{v|v=x_1v_1+x_2v_2+\dots+x_kv_k\,\ x_1,x_2\dots,x_k\in R\}\).

We also say that \(\operatorname{Span}(v_1,v_2,\dots v_k)\) is the subset spanned by generated by the vectors \(v_1,v_2,\dots,v_k\).

We can easily determine whether one vector is in the span of some vectors or not, the only we should do is to find whether the linear equations have solution.

Try to research the dimensions of the span of several vectors:

In a xy-plane, it is easy to discover, when several vectors are collinear vectors, its spans is that line that they both belong to (1-dimentional), otherwise, it is the whole xy-coordinate (2-dimentional).

In a xyz-space, when several vectors are collinear vectors, its spans is that line that they both belong to (1-dimentional), when several vectors are coplanar vectors, its spans is the plane that they both belong to (2-dimentional), otherwise, it is the whole xyz-coordinate (3-dimentional).

Expend it to higher dimensions space, we can get a conclusion of the dimensions in spans, the dimention of the spans is the rank of the matrix made up by the vectors.

Special thanks

Referance Material: \(\texttt{Interactive Linear Algebra}\)