VECTORS · TEXTBOOK LESSON
Linear algebra studies vectors, matrices and linear transformations. It provides a compact language for solving systems, representing geometric transformations and modelling multivariable relationships.
Learning objectives
By the end of this lesson, you should be able to:
- Perform valid vector or matrix operations.
- Interpret dimensions and compatibility of matrices.
- Use matrices to represent systems or transformations.
- Check results using structural properties such as dimensions or substitution.
1. Key vocabulary and definition
Definition
A matrix is a rectangular array of entries. A vector can be represented as an ordered list or column matrix. Matrix multiplication is defined when the inner dimensions agree.
vectormatrixdimensiondeterminantlinear transformationsystem
2. Essential facts and rules
These are the facts you should know before attempting the worked example.
Key mathematical facts
- An m×n matrix has m rows and n columns.
- Matrix multiplication is generally not commutative: AB may not equal BA.
- For a 2×2 matrix [[a,b],[c,d]], determinant = ad-bc.
3. Standard method
A reliable method helps prevent errors and makes your reasoning easy to follow.
1
Write dimensions before operating on matrices.
2
For addition, combine corresponding entries.
3
For multiplication, take row-by-column dot products.
4
For systems, represent coefficients and unknowns consistently.
5
Check by multiplying back or substituting into the original equations.
4. Worked example
Worked example
Multiply A=[[1,2],[3,4]] by vector v=[[5],[6]].
1
First entry: 1×5 + 2×6 = 17.
2
Second entry: 3×5 + 4×6 = 39.
3
Therefore Av = [[17],[39]].
4
The dimensions are (2×2)(2×1)→(2×1), so the multiplication is valid.
5. Common mistakes
Watch out for these
- Starting calculations before deciding what the question is asking.
- Skipping important steps or changing notation midway through a solution.
- Accepting an answer without checking whether its size, sign or unit is sensible.
6. Video lesson
This topic is ready for a matching video lesson. The written textbook lesson is fully available now; additional videos can be added to the same lesson template.
7. Practice exercise
Complete these without looking at the answers first.
- Add [[1,2],[3,4]] and [[5,6],[7,8]].
- Find the determinant of [[2,1],[5,3]].
- Compute the dot product (1,2,3)·(4,0,-1).
- State the dimension of a matrix with 3 rows and 5 columns.
- Explain why a 2×3 matrix cannot be added to a 3×2 matrix.
Show answers
- [[6,8],[10,12]]
- 1
- 1
- 3×5
- Matrix addition requires equal dimensions
8. Chapter summary
- Understand the underlying idea before memorising a procedure.
- Use precise mathematical notation and show a logical method.
- Always check the final result.