§5 Vector Spaces

1. The Big Idea

Vector spaces are about building objects using addition and scalar multiplication.

The objects might be ordinary vectors, polynomials, matrices, or something more abstract. The same questions keep coming back:

1. Can these objects build the whole space?
2. Is any object redundant?
3. Do we have the perfect set of building blocks?

Here is the vocabulary in plain language:

The main theme is simple: a basis is the cleanest possible way to describe a space.

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2. Linear Combinations

A vector $\vec{w}$ is a linear combination of vectors $\vec{v}_1,\vec{v}_2,\dots,\vec{v}_r$ if

$$\vec{w}=a_1\vec{v}_1+a_2\vec{v}_2+\cdots+a_r\vec{v}_r$$

where $a_1,a_2,\dots,a_r$ are scalars.

The scalars are called coefficients.

Think of a linear combination as a recipe:

• the vectors are the ingredients
• the coefficients say how much of each ingredient to use
• the final result is the vector you build

For example, if

$$\vec{v}_1=(1,0),\qquad \vec{v}_2=(0,1)$$

then

$$3\vec{v}_1+5\vec{v}_2=(3,5)$$

So $(3,5)$ is a linear combination of $\vec{v}_1$ and $\vec{v}_2$.

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3. Span

The span of a set of vectors is the set of all possible linear combinations of those vectors.

$$\operatorname{span}\{\vec{v}_1,\vec{v}_2,\dots,\vec{v}_r\}$$

means all vectors you can build using $\vec{v}_1,\vec{v}_2,\dots,\vec{v}_r$.

If a set spans a space, then the vectors in the set can build every vector in that space.

Standard Basis of $\mathbb R^3$

Let

$$e_1=(1,0,0),\qquad e_2=(0,1,0),\qquad e_3=(0,0,1)$$

These vectors span $\mathbb R^3$, because every vector

$$(x,y,z)$$

can be written as

$$(x,y,z)=x(1,0,0)+y(0,1,0)+z(0,0,1)$$

So $e_1,e_2,e_3$ can build every vector in $\mathbb R^3$.

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4. Spanning Polynomial Spaces

Vector spaces do not have to contain arrows. They can contain polynomials too.

For $P_2$, the space of polynomials of degree at most 2, every polynomial looks like

$$a+bx+cx^2$$

The standard spanning set is

$$\{1,x,x^2\}$$

because

$$a+bx+cx^2=a(1)+b(x)+c(x^2)$$

This is the same idea as vectors in $\mathbb R^3$. The objects look different, but we are still building everything from a few basic pieces.

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5. Spanning Matrix Spaces

Matrices can also form vector spaces.

For $M_{2\times 2}$, the space of all $2\times 2$ matrices, every matrix looks like

$$\begin{bmatrix} a&b\\ c&d \end{bmatrix}$$

The standard spanning set is

$$\left\{ \begin{bmatrix} 1&0\\ 0&0 \end{bmatrix}, \begin{bmatrix} 0&1\\ 0&0 \end{bmatrix}, \begin{bmatrix} 0&0\\ 1&0 \end{bmatrix}, \begin{bmatrix} 0&0\\ 0&1 \end{bmatrix} \right\}$$

Each matrix turns on one entry and leaves the others zero.

Any matrix in $M_{2\times 2}$ can be built as

$$\begin{bmatrix} a&b\\ c&d \end{bmatrix} = a\begin{bmatrix} 1&0\\ 0&0 \end{bmatrix} + b\begin{bmatrix} 0&1\\ 0&0 \end{bmatrix} + c\begin{bmatrix} 0&0\\ 1&0 \end{bmatrix} + d\begin{bmatrix} 0&0\\ 0&1 \end{bmatrix}$$

So those four matrices span $M_{2\times 2}$.

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6. How to Check Linear Combinations and Span

To check whether a vector is in a span, turn the question into a system of equations.

Suppose you want to know whether $\vec{w}$ is a linear combination of $\vec{v}_1,\vec{v}_2,\vec{v}_3$.

Set up

$$\vec{w}=a\vec{v}_1+b\vec{v}_2+c\vec{v}_3$$

Then solve for $a,b,c$.

Checking Whether Vectors Span $\mathbb R^n$

If you have $n$ vectors in $\mathbb R^n$, place them as columns in a matrix:

$$A= \begin{bmatrix} |&|&|\\ \vec{v}_1&\vec{v}_2&\vec{v}_3\\ |&|&| \end{bmatrix}$$

For three vectors in $\mathbb R^3$:

$$\det(A)\neq 0$$

means the vectors span $\mathbb R^3$.

If

$$\det(A)=0$$

then they do not span all of $\mathbb R^3$.

This determinant shortcut works when the number of vectors matches the dimension, so the matrix is square.

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7. Linear Independence

A set of vectors is linearly independent if no vector in the set is redundant.

The formal test is:

$$a_1\vec{v}_1+a_2\vec{v}_2+\cdots+a_r\vec{v}_r=\vec{0}$$

If the only solution is

$$a_1=a_2=\cdots=a_r=0$$

then the vectors are linearly independent.

This is called the trivial solution.

If there is a solution where at least one coefficient is not zero, then the vectors are linearly dependent.

Meaning

So linear independence is really a no-redundancy test.

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8. Fast Independence Test with Determinants

For $n$ vectors in $\mathbb R^n$, place the vectors as columns in a square matrix $A$.

If

$$\det(A)\neq 0$$

then the vectors are linearly independent.

If

$$\det(A)=0$$

then the vectors are linearly dependent.

This is one of the fastest exam methods, but remember the condition: it works directly when you have exactly $n$ vectors in $\mathbb R^n$.

For example, three vectors in $\mathbb R^3$ form a $3\times 3$ matrix, so the determinant test applies.

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9. Basis

A basis is a set of vectors that does two things:

1. It spans the space.
2. It is linearly independent.

So a basis is the perfect set of building blocks.

It has enough vectors to build the whole space, but no extra unnecessary vectors.

Basis of $\mathbb R^3$

The standard basis is

$$\{(1,0,0),(0,1,0),(0,0,1)\}$$

Every vector in $\mathbb R^3$ can be built from these vectors, and none of them is redundant.

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10. Coordinates Relative to a Basis

Coordinates depend on the basis you choose.

If

$$S=\{\vec{v}_1,\vec{v}_2,\dots,\vec{v}_n\}$$

is an ordered basis, and

$$\vec{v}=c_1\vec{v}_1+c_2\vec{v}_2+\cdots+c_n\vec{v}_n$$

then the coordinate vector of $\vec{v}$ relative to basis $S$ is

$$[\vec{v}]_S= \begin{bmatrix} c_1\\ c_2\\ \vdots\\ c_n \end{bmatrix}$$

These coordinates are not the vector itself. They are the instructions for building the vector from the basis $S$.

The order of the basis matters. If you swap the order of the basis vectors, the coordinate vector changes too.

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11. Basis for Subspaces

A subspace is a smaller vector space sitting inside a larger vector space.

To find a basis for a subspace, look for the free parameters. Each free parameter usually gives one basis vector.

Example: Trace-Zero $2\times 2$ Matrices

Let

$$M=\left\{ A\in M_{2\times 2}:\operatorname{tr}(A)=0 \right\}$$

This means $M$ is the set of all $2\times 2$ matrices whose trace is zero.

A general matrix in $M_{2\times 2}$ looks like

$$A= \begin{bmatrix} a&b\\ c&d \end{bmatrix}$$

The trace condition is

$$a+d=0$$

so

$$d=-a$$

Thus every matrix in this subspace looks like

$$A= \begin{bmatrix} a&b\\ c&-a \end{bmatrix}$$

Now break it apart by parameters:

$$A = a \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix} + b \begin{bmatrix} 0&1\\ 0&0 \end{bmatrix} + c \begin{bmatrix} 0&0\\ 1&0 \end{bmatrix}$$

So a basis is

$$\left\{ \begin{bmatrix} 1&0\\ 0&-1 \end{bmatrix}, \begin{bmatrix} 0&1\\ 0&0 \end{bmatrix}, \begin{bmatrix} 0&0\\ 1&0 \end{bmatrix} \right\}$$

There are three free parameters: $a$, $b$, and $c$. That is why the basis has three matrices and the subspace has dimension 3.

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12. Problem-Solving Routine

For vector space problems, use this routine:

1. Identify the space. Are you working in $\mathbb R^n$, a polynomial space, a matrix space, or a subspace?
2. To test a linear combination, set up coefficients. Write the target as a combination of the given vectors and solve.
3. To test span, ask whether every object can be built. For $n$ vectors in $\mathbb R^n$, use a determinant if possible.
4. To test independence, set the combination equal to zero. Only the trivial solution means independent.
5. To find a basis, keep enough vectors to span but remove redundancy. A basis must do both jobs.
6. For subspaces, look for free parameters. Break the general object into parameter times basic objects.
7. Dimension equals the number of basis vectors. Count the final basis elements.

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13. Key Takeaways

• A linear combination is a recipe made from scalar multiples and addition.
• The span is everything those recipes can build.
• A vector is in a span if the matching coefficient system has a solution.
• Linear independence means no redundancy.
• A basis is both spanning and linearly independent.
• Coordinates relative to a basis are instructions for building a vector from that basis.
• Dimension is the number of vectors in a basis.
• For subspaces, free parameters often reveal the basis.

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Mini-Self-Check

1. What does it mean for $\vec{w}$ to be in $\operatorname{span}\{\vec{v}_1,\vec{v}_2\}$?

2. What is the difference between a spanning set and a basis?

3. If three vectors in $\mathbb R^3$ are placed as columns of $A$ and $\det(A)=0$, what does that mean?

4. Why does order matter in a basis?

5. Why is the trace-zero $2\times 2$ matrix subspace dimension 3?