Subspace and linear independence

Vector subspace

Vector space: a set of elements that can be added together and multiplied by scalars (for this class, in $\mathbb{R}$ or $\mathbb{C}$). Vectors and matrices

Example: ${\mathbb{R}}^n$.

A subspace of ${\mathbb{R}}^n$ is a vector space that is a subset of ${\mathbb{R}}^n$.

Example: take $k$ vectors $x_1$, …, $x_k$, you can check that all vectors of the form

$$a_1{x}_1+\cdots +a_k{x}_k$$

form a subspace. We will denote this subspace as $S=\text{span}\{x_1,\dots ,x_k\}$.

Linear independence

We say that $x_1$, …, $x_k$ are linearly independent if and only if any vector $x\in S$ has a unique decomposition as

$$x=a_1{x}_1+\cdots +a_k{x}_k$$

In particular, if

$$0=a_1{x}_1+\cdots +a_k{x}_k$$

then $a_1=\cdots =a_k=0$.

If the vectors are linearly independent, the dimension of $S$ is equal to $k$.

Linear independence will be important when solving linear systems. It will guarantee that the solution is unique.

Direct sum

If $U$ and $V$ are subspaces, then $U+V$ is a subspace. We say that $W=U\oplus V$ is the direct sum of $U$ and $V$ if $U\cap V=\{0\}$. The direct sum means that if a vector is decomposed into its $U$ and $V$ components, this decomposition is unique.

Example: verify that if $x_1$, …, $x_k$ are linearly independent, then

$$S=\text{span}\{x_1\}\oplus \cdots \oplus \text{span}\{x_k\}$$