Constructing Bases and Coordinate Representations

students, imagine trying to describe a movie scene using only a few well-chosen directions 🎬. Instead of repeating every detail, you use a small set of “building blocks” that can recreate the whole scene. In linear algebra, that idea shows up in bases and coordinate representations. A basis gives us a minimal set of vectors that can build every vector in a space, and coordinates tell us exactly how to combine those basis vectors.

What a Basis Does

A span is the set of all linear combinations of some vectors. If vectors can generate every vector in a space, then they span that space. But spanning alone is not enough if we want a clean, efficient description. We also want the vectors to be linearly independent, meaning none of them is unnecessary.

A basis is a set of vectors that satisfies both conditions:

it spans the space,
it is linearly independent.

This makes a basis a perfect toolkit 🧰. Every vector in the space can be written in exactly one way as a linear combination of the basis vectors.

For example, in $\mathbb{R}^2$, the vectors $\begin{bmatrix}1\\0\end{bmatrix}$ and $\begin{bmatrix}0\\1\end{bmatrix}$ form a basis. Any vector $\begin{bmatrix}x\y\end{bmatrix}$ can be written as

$$\begin{bmatrix}x\y\end{bmatrix}=x\begin{bmatrix}1\\0\end{bmatrix}+y\begin{bmatrix}0\\1\end{bmatrix}.$$

That means the coordinates of the vector relative to this basis are simply $(x,y)$.

Why Coordinate Representations Matter

A coordinate representation tells us how to express a vector using a chosen basis. If $B=\{\mathbf{b}_1,\mathbf{b}_2,\dots,\mathbf{b}_n\}$ is a basis, then any vector $\mathbf{v}$ in the space can be written as

$$\mathbf{v}=c_1\mathbf{b}_1+c_2\mathbf{b}_2+\cdots+c_n\mathbf{b}_n.$$

The numbers $c_1,c_2,\dots,c_n$ are called the coordinates of $\mathbf{v}$ with respect to $B$.

These coordinates are like instructions for rebuilding the vector from the basis. 📦 If the basis changes, the coordinate values usually change too, even though the vector itself stays the same.

A Simple Example

Let $B=\left\{\begin{bmatrix}1\\1\end{bmatrix},\begin{bmatrix}1\\-1\end{bmatrix}\right\}$ in $\mathbb{R}^2$. Suppose we want to find the coordinates of $\mathbf{v}=\begin{bmatrix}3\\1\end{bmatrix}$ with respect to $B$.

We want numbers $c_1$ and $c_2$ such that

$$c_1\begin{bmatrix}1\\1\end{bmatrix}+c_2\begin{bmatrix}1\\-1\end{bmatrix}=\begin{bmatrix}3\\1\end{bmatrix}.$$

This gives the system

$$c_1+c_2=3$$

$$c_1-c_2=1.$$

Adding the equations gives $2c_1=4$, so $c_1=2$. Then $c_2=1$. So the coordinate vector of $\mathbf{v}$ relative to $B$ is

$$[\mathbf{v}]_B=\begin{bmatrix}2\\1\end{bmatrix}.$$

This means

$$\mathbf{v}=2\begin{bmatrix}1\\1\end{bmatrix}+1\begin{bmatrix}1\\-1\end{bmatrix}.$$

Constructing a Basis from a Spanning Set

Sometimes you start with a set of vectors that spans a space, but the set includes redundant vectors. To construct a basis, you remove the extra vectors while keeping the span the same.

A common method is:

Put the vectors into a matrix as columns.
Row reduce the matrix.
Identify the pivot columns.
Keep the original vectors corresponding to the pivot columns.

Why does this work? The pivot columns show which vectors add new directions to the span. Non-pivot columns can be written as combinations of earlier pivot columns, so they are unnecessary.

Example: Finding a Basis from Vectors in $\mathbb{R}^3$

Consider the vectors

$$\mathbf{v}_1=\begin{bmatrix}1\\0\\1\end{bmatrix},\quad \mathbf{v}_2=\begin{bmatrix}2\\1\\3\end{bmatrix},\quad \mathbf{v}_3=\begin{bmatrix}3\\1\\4\end{bmatrix}.$$

Place them in a matrix:

$$A=\begin{bmatrix}1&2&3\\0&1&1\\1&3&4\end{bmatrix}.$$

Row reducing gives pivot columns $1$ and $2$; column $3$ is dependent on the first two. So a basis for the span is

$$\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}2\\1\\3\end{bmatrix}\right\}.$$

This means every vector in the span of the original set can be built using just those two vectors.

From Linear Independence to Uniqueness

Linear independence is the reason coordinate representations are unique. If a basis were not linearly independent, then one vector could be written using the others, and different coordinate choices could produce the same vector.

Suppose

$$c_1\mathbf{b}_1+c_2\mathbf{b}_2+\cdots+c_n\mathbf{b}_n=\mathbf{0}.$$

If the only solution is

$$c_1=c_2=\cdots=c_n=0,$$

the set is linearly independent.

Now imagine two different coordinate representations for the same vector:

$$\mathbf{v}=c_1\mathbf{b}_1+\cdots+c_n\mathbf{b}_n$$

and

$$\mathbf{v}=d_1\mathbf{b}_1+\cdots+d_n\mathbf{b}_n.$$

Subtracting gives

$$0=(c_1-d_1)\mathbf{b}_1+\cdots+(c_n-d_n)\mathbf{b}_n.$$

If the basis vectors are linearly independent, then each difference must be zero, so $c_i=d_i$ for every $i$. That proves uniqueness.

This is a major reason bases are so useful: they make representation both possible and exact. ✅

Coordinate Vectors and Matrix Form

If $B=\{\mathbf{b}_1,\dots,\mathbf{b}_n\}$ is a basis of a space, we can form a matrix

$$P_B=\begin{bmatrix}\mathbf{b}_1&\mathbf{b}_2&\cdots&\mathbf{b}_n\end{bmatrix}.$$

Then the coordinate relation becomes

$$P_B[\mathbf{v}]_B=\mathbf{v}.$$

This means the coordinate vector $[\mathbf{v}]_B$ is transformed into the actual vector $\mathbf{v}$ by multiplying with the basis matrix.

If $P_B$ is invertible, then

$$[\mathbf{v}]_B=P_B^{-1}\mathbf{v}.$$

This formula is powerful because it gives a direct way to compute coordinates once the basis matrix is known.

Example with a Nonstandard Basis

Let

$$B=\left\{\begin{bmatrix}1\\2\end{bmatrix},\begin{bmatrix}3\\1\end{bmatrix}\right\}.$$

Find the coordinates of $\mathbf{v}=\begin{bmatrix}7\\5\end{bmatrix}$.

Set up

$$\begin{bmatrix}1&3\\2&1\end{bmatrix}\begin{bmatrix}c_1\c_2\end{bmatrix}=\begin{bmatrix}7\\5\end{bmatrix}.$$

This gives

$$c_1+3c_2=7$$

$$2c_1+c_2=5.$$

Solving, we get $c_2=\frac{9}{5}$ and $c_1=\frac{8}{5}$. Therefore,

$$[\mathbf{v}]_B=\begin{bmatrix}\frac{8}{5}\\frac{9}{5}\end{bmatrix}.$$

Even though the coordinates are fractions, they still exactly describe how to build $\mathbf{v}$ from the basis vectors.

Dimension and Why Bases Have the Same Size

The dimension of a vector space is the number of vectors in any basis of that space. This works because all bases of a given space have the same number of vectors.

For example:

the dimension of $\mathbb{R}^2$ is $2$,
the dimension of $\mathbb{R}^3$ is $3$,
the space of all $2\times 2$ matrices has dimension $4$.

A basis is not just any spanning set; it is the smallest kind of spanning set possible. If a space has dimension $n$, then every basis must contain exactly $n$ vectors.

This helps in real situations. For instance, in computer graphics and robotics, coordinates relative to a special basis can make calculations easier. A rotated coordinate system may align with a shape, making motion or measurement simpler to handle.

Conclusion

Constructing bases and coordinate representations is about turning complicated vector descriptions into organized, reusable building blocks 🌟. A basis is a spanning, linearly independent set, and every vector in the space has one and only one coordinate representation with respect to that basis. To construct a basis from a spanning set, remove dependent vectors until only the essential ones remain. Then use those basis vectors to express any vector as a unique linear combination. This connects directly to span, linear independence, and dimension, which are central ideas in linear algebra.

Study Notes

A span is all linear combinations of a set of vectors.
A basis is a set of vectors that both spans a space and is linearly independent.
A basis gives a unique way to represent every vector in the space.
The coordinates of a vector relative to a basis are the coefficients in its linear combination form.
To construct a basis from a spanning set, remove dependent vectors while keeping the span unchanged.
Row reduction and pivot columns help identify which vectors belong in a basis.
If $B=\{\mathbf{b}_1,\dots,\mathbf{b}_n\}$, then any vector $\mathbf{v}$ can be written as $\mathbf{v}=c_1\mathbf{b}_1+\cdots+c_n\mathbf{b}_n$.
The coordinate vector is written as $[\mathbf{v}]_B$.
The matrix $P_B=\begin{bmatrix}\mathbf{b}_1&\cdots&\mathbf{b}_n\end{bmatrix}$ links coordinates to actual vectors.
If $P_B$ is invertible, then $[\mathbf{v}]_B=P_B^{-1}\mathbf{v}$.
The dimension of a space is the number of vectors in any basis of that space.