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tags: Giang's linear algebra notes
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# Chapter 5: Coordinate changes

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## Coordinate changes

**Change of basis**: Let $V$ be a vector space and $\beta,\beta'$ are two ordered bases. Let $v\in V$ be a vector in $V$, then the relationship between $[v]^{\beta}$ and $[v]^{\beta'}$ is
$$[v]^{\beta'}=[I_{V}]_{\beta}^{\beta'}[v]^{\beta}$$
where $I_{V}:V\to V$ is the identity operator, i.e. $I_{V}(v)=v$.

**Change-of-coordinate matrix**: The matrix $[I_{V}]_{\beta}^{\beta'}$ in theorem above is called *the change-of-coordinate matrix from $\beta$ to $\beta'$*, i.e. the matrix we use when we want to convert $\beta$ to $\beta'$

>**NOTE**: $[I_{V}]_{\beta}^{\beta'}$ measures how much of $\beta'$ lies in $\beta$

>**NOTE**: Change-of-coordinate matrices are always square and always invertible

## Coordinate change and matrices

**Coordinate change and matrices**: Let $V$ be a vector space with two bases $\beta'$ and $\beta$, and let $Q=[I_{V}]_{\beta}^{\beta'}$ be change-of-coordinate matrix from $\beta$ to $\beta'$. Let $T:V\to V$ be a linear transformation, then the relationship between $[T]_{\beta}^{\beta}$ and $[T]_{\beta}^{\beta'}$ is
$$[T]_{\beta}^{\beta'}=Q\cdot[T]_{\beta}^{\beta}\cdot Q^{-1}$$

**Matrix similarity**: Two $n\times n$ matrices $A,B$ are said to be similar if one has
$B=QAQ^{-1}$ for some invertible matrix $Q$

>**NOTE**: Matrix similarity is an important notion in linear algebra

## Expansions

**Automorphisms**. Transformations mapping onto themselves are called *automorphisms*