Why this matters
Every model you will build, from a simple price predictor to a language model, stores its data and its learned settings as vectors and matrices. Job adverts for data science roles in Accra name linear algebra for exactly this reason. Once you can read these objects, error messages like "shapes do not match" stop being scary and start being useful clues.
Vectors
A vector is an ordered list of numbers. One farm's record can be a vector: rainfall of 5 (hundreds of millimetres) and 2 bags of fertiliser gives [5, 2]. Order matters: [2, 5] would mean something different. The number of entries is the vector's length or dimension.
The most important operation is the dot product. Multiply matching entries and add the results. With weights [1.5, 0.8]:
[5, 2] · [1.5, 0.8] = 5 × 1.5 + 2 × 0.8 = 7.5 + 1.6 = 9.1
That is exactly what a simple model does: each input is multiplied by its weight and the results are added. Add a bias of 1 and the predicted yield is 10.1 bags. A weight shows how much the model cares about each input.
Matrices
A matrix is a grid of numbers with rows and columns. Stack several farm vectors as rows and you get a matrix where each row is a farm and each column is a feature. Three farms with two features gives a matrix of (3, 2): three rows, two columns.