Linear Algebra/Basis

We denote a basis with angle brackets $⟨\overrightarrow{{\beta }_1},\overrightarrow{{\beta }_2},\dots ⟩$ to signify that this collection is a sequence — the order of the elements is significant. (The requirement that a basis be ordered will be needed, for instance, in Definition 1.13.)

(Calculus books refer to ${ℝ}^2$'s standard basis vectors $\overrightarrow{ı}$ and $\overrightarrow{ȷ}$ instead of ${\overrightarrow{e}}_1$ and ${\overrightarrow{e}}_2$, and they refer to ${ℝ}^3$'s standard basis vectors $\overrightarrow{ı}$, $\overrightarrow{ȷ}$, and $\overrightarrow{k}$ instead of ${\overrightarrow{e}}_1$, ${\overrightarrow{e}}_2$, and ${\overrightarrow{e}}_3$.) Note that the symbol "${\overrightarrow{e}}_1$" means something different in a discussion of ${ℝ}^3$ than it means in a discussion of ${ℝ}^2$.

Consider again Example 1.2. It involves two verifications.

In the first, to check that the set is linearly independent we looked at linear combinations of the set's members that total to the zero vector $c_1{\overrightarrow{\beta }}_1+c_2{\overrightarrow{\beta }}_2=(\genfrac{}{}{0ex}{}{0}{0})$. The resulting calculation shows that such a combination is unique, that $c_1$ must be $0$ and $c_2$ must be $0$.

The second verification, that the set spans the space, looks at linear combinations that total to any member of the space $c_1{\overrightarrow{\beta }}_1+c_2{\overrightarrow{\beta }}_2=(\genfrac{}{}{0ex}{}{x}{y})$. In Example 1.2 we noted only that the resulting calculation shows that such a combination exists, that for each $x,y$ there is a $c_1,c_2$. However, in fact the calculation also shows that the combination is unique: $c_1$ must be $(y-x)/2$ and $c_2$ must be $2x-y$.

That is, the first calculation is a special case of the second. The next result says that this holds in general for a spanning set: the combination totaling to the zero vector is unique if and only if the combination totaling to any vector is unique.

We consider combinations to be the same if they differ only in the order of summands or in the addition or deletion of terms of the form "$0\cdot \overrightarrow{\beta }$".

We will later do representations in contexts that involve more than one basis. To help with the bookkeeping, we shall often attach a subscript $B$ to the column vector.

Our main use of representations will come in the third chapter. The definition appears here because the fact that every vector is a linear combination of basis vectors in a unique way is a crucial property of bases, and also to help make two points. First, we fix an order for the elements of a basis so that coordinates can be stated in that order. Second, for calculation of coordinates, among other things, we shall restrict our attention to spaces with bases having only finitely many elements. We will see that in the next subsection.

Exercises

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