Chapter practice · Linear Algebra

Intuition

Bring the chapter together: read and classify a matrix, add and scale, apply one to a vector, decide whether a map is linear, build a matrix from where the standard vectors go, multiply, invert and transpose. No worked example sits above the answers.

Following a route means knowing which turn belongs to which junction. The chapter's ideas differ in the same way: each question is really asking which one applies.

Before you practise

Decide first what is being asked. Shape questions count rows then columns; a product with a vector weights columns; linearity is two conditions; the matrix of a map is built from images of the standard vectors; a product of matrices composes right to left, and each of its entries pairs a row with a column; a transpose turns rows into columns.

$A\mathbf{x}=x_1\mathbf{a}_1+\cdots+x_n\mathbf{a}_n$ : weights on columns.

Reminders

Rows first in a shape, and row first in $a_{ij}$ .
Column j of a matrix is the image of $\mathbf{e}_j$ under its map.
Linearity needs both conditions, and a map with $T\mathbf{0}\neq\mathbf{0}$ fails at once.

Applications