Dimension

Section 4.4 Dimension

Goals.

We will

prove that coordinates for vectors are unique in any vector space, and
that all bases for one vector space have the same size.

Subsection 4.4.1 Uniqueness

Our first task is to show that for any vector there is only one coordinate that works to generate that vector.

Example 4.4.1.

We will show that only one coordinate works by supposing there are two and ending up with those two being equal.

Suppose \(\vec{x}=a_1\vec{b}_1+a_2\vec{b}_2+a_3\vec{b}_3\) and \(\vec{x}=c_1\vec{b}_1+c_2\vec{b}_2+c_3\vec{b}_3\text{.}\)

(a)

Set the linear combinations equal to each other.

Solution.

\begin{equation*} a_1\vec{b}_1+a_2\vec{b}_2+a_3\vec{b}_3 = c_1\vec{b}_1+c_2\vec{b}_2+c_3\vec{b}_3. \end{equation*}

(b)

Collect like terms (\(\vec{b}_1,\) \(\vec{b}_2,\) \(\vec{b}_3\) are the terms) on one side.

Solution.

\begin{align*} a_1\vec{b}_1+a_2\vec{b}_2+a_3\vec{b}_3 = & c_1\vec{b}_1+c_2\vec{b}_2+c_3\vec{b}_3.\\ (a_1-c_1)\vec{b}_1+(a_2-c_2)\vec{b}_2+(a_3-c_3)\vec{b}_3 = & \vec{0}. \end{align*}

(c)

What does this equation imply about the coefficients?

Solution.

Because there is a zero on the right, each coefficient must be zero.

\begin{align*} a_1-c_1 = & 0.\\ a_2-c_2 = & 0.\\ a_3-c_3 = & 0.\\ a_1 = & c_1.\\ a_2 = & c_2.\\ a_3 = & c_3. \end{align*}

(d)

What does this answer imply about how many, distinct coordinates a single vector possesses with respect to one basis?

Subsection 4.4.2 Dimension

Now that we know that each vector has a unique coordinate we can begin to learn when we can analyze coordinates rather than the original vectors. First we ask if a set of independent vectors has independent coordinates.

Activity 4.4.1.

Note \(\{x^2+x+1,x+1,1\}\) is a linearly independent set.

(a)

How many solutions does \(a_1(x^2+x+1)+a_2(x+1)+\ldots+a_n(1)=\vec{0}\) have?

(b)

Find the coordinates for each of these vectors with respect to the standard basis \(\B=\{1,x,x^2\}\text{.}\)

(c)

Check if these coordinates (the coordinates are now the vectors) are linearly dependent or independent.

(d)

Would this always be true?

Activity 4.4.2.

We will determine if the coordinates of this set of dependent vectors are independent.\(\{x^2+x+1,x^2+x,x+1,1\}\)

(a)

Find the coordinates for each vector with respect to the standard basis \(\B=\{1,x,x^2\}\text{.}\)

(b)

Check if the coordinates are linearly dependent or independent.

(c)

In how many dimensions do the coordinates exist? Call this \(m\text{.}\)

(d)

In \(\R^m\) how many vectors can be linearly independent?

(e)

What does this imply about the original vectors?

We have illustrated that the coordinates of an independent or dependent set are also independent or dependent. This suggests that we can work with the coordinates rather than the original vectors.

Now we are ready to address the question about the size of all bases for a vector space. Suppose one basis is size \(n=3\) and another is size \(m=4\text{.}\) The vectors in the basis of size 4 have coordinates in terms of the basis of size 3. But in that basis any set of size four has to have at least one dependent set. Thus there cannot be a basis of size 3 and 4. In general if \(\{\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_n\}\) and \(\{\vec{w}_1,\vec{w}_2,\ldots,\vec{w}_m\}\) are both bases \(n=m\text{.}\)

Definition 4.4.2.

For any basis \(\B\) of a vector space \(V\) the size of a basis \(|\B|\) is called the dimension of \(V\text{.}\)

This leads to one last question. If \(C=\{\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_n\}\) is a linearly independent set but \(\mbox{span}(C)\) is not the whole vector space, can it be extended into a basis? The answer is yes and will be shown later in the course.

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