## Commutative and distributive properties for vector inner products

As I think I’ve previously mentioned, one of the minor problems with Gilbert Strang’s book Linear Algebra and Its Applications, Third Edition, is that frequently Strang will gloss over things that in a more rigorous treatment really should be explicitly proved.

In particular, Strang seems to assume and then subsequently use (e.g., in his proof of the Schwarz inequality) the following commutative and distributive properties for vector inner products:

$x^Ty = y^Tx$

$x^T(y+z) = x^Ty + x^Tz$

The proofs of these are simple, and rely on the commutative and distributive properties of standard addition and multiplication for real numbers:

Let $x$, $y$ and $z$ be vectors in $\mathbb{R}^n$, so that $x = \left(x_1, \dots, x_n\right)$, $y = \left(y_1, \dots, y_n\right)$ and $z = \left(z_1, \dots, z_n\right)$ where all $x_i$, $y_i$ and $z_i$ are real numbers.

By the definition of the inner product and the commutative property of standard multiplication for real numbers we have

$x^Ty = \sum_{i=1}^n x_iy_i = \sum_{i=1}^n y_ix_i = y^Tx$

By the definition of the inner product and the distributive property of standard addition and multiplication for real numbers we have

$x^T(y+z) = \sum_{i=1}^n x_i\left(y_i+z_i\right)$

$= \sum_{i=1}^n \left(x_iy_i+x_iz_i\right)$

$= \sum_{i=1}^n x_iy_i + \sum_{i=1}^n x_iz_i = x^Ty + x^Tz$

Note that by combining the commutative and distributive properties proved above we also have

$\left(y+z\right)^Tx = x^T\left(y+z\right)$

$x^Ty + x^Tz = y^Tx + z^Tx$

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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