4.1 Vector spaces and linear maps

A vector space $V$ over a field $k$ is a set with a consistent way to take linear combinations of elements with coefficients in $k$ . We will only be using the cases $k = R$ and $k = C$ , so such finite dimensional $V$ will just be $R^{n}$ or $C^{n}$ . Choosing a basis (set of $n$ linearly independent vectors) ${e_{j}}$ , an arbitrary vector $v \in V$ can be written as

v = v_{1} e_{1} + v_{2} e_{2} + \dots + v_{n} e_{n}

giving an explicit identification of $V$ with $n$ -tuples $v_{j}$ of real or complex numbers which we will usually write as column vectors

v = v_{1} v_{2} ⋮ v_{n}

Then the expansion of a vector $v$ with respect to the basis can be written

v = (e_{1} e_{2} \dots e_{n}) v_{1} v_{2} ⋮ v_{n}

The choice of a basis ${e_{j}}$ also allows us to express the action of a linear operator $L$ on $V$

L : v \in V \to Lv \in V

as multiplication by an $n$ by $n$ matrix:

v_{1} v_{2} ⋮ v_{n} \to L_{11} L_{21} ⋮ L_{n 1} L_{12} L_{22} ⋮ L_{n 2} \dots \dots ⋮ \dots L_{1 n} L_{2 n} ⋮ L_{nn} v_{1} v_{2} ⋮ v_{n}

The reader should be warned that we will often not notationally distinguish between a linear operator $L$ and its matrix with matrix entries $L_{j k}$ with respect to some unspecified basis, since we are often interested in properties of operators $L$ that, for the corresponding matrix, are basis-independent (e.g., is the operator or matrix invertible?). The invertible linear operators on $V$ form a group under composition, a group we will sometimes denote $G L (V)$ , with “GL” indicating “General Linear”. Choosing a basis identifies this group with the group of invertible matrices, with group law matrix multiplication. For $V$ $n$ dimensional, we will denote this group by $G L (n, R)$ in the real case, $G L (n, C)$ in the complex case.

Note that when working with vectors as linear combinations of basis vectors, we can use matrix notation to write a linear transformation as

v \to Lv = (e_{1} e_{2} \dots e_{n}) L_{11} L_{21} ⋮ L_{n 1} L_{12} L_{22} ⋮ L_{n 2} \dots \dots ⋮ \dots L_{1 n} L_{2 n} ⋮ L_{nn} v_{1} v_{2} ⋮ v_{n}

We see from this that we can think of the transformed vector as we did above in terms of transformed coefficients $v_{j}$ with respect to fixed basis vectors, but also could leave the $v_{j}$ unchanged and transform the basis vectors. At times we will want to use matrix notation to write formulas for how the basis vectors transform in this way, and then will write

e_{1} e_{2} ⋮ e_{n} \to L_{11} L_{12} ⋮ L_{1 n} L_{21} L_{22} ⋮ L_{2 n} \dots \dots ⋮ \dots L_{n 1} L_{n 2} ⋮ L_{nn} e_{1} e_{2} ⋮ e_{n}

Note that putting the basis vectors $e_{j}$ in a column vector like this causes the matrix for $L$ to act on them by the transposed matrix.

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正文：英文 · 原著转录

核对状态：AI 辅助转录核对，未作人工审阅

原书 PDF · 印刷页 34、35

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

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书架

4.1 Vector spaces and linear maps