LINEAR-ALGEBRA · Unit 10 · Video 3 · Interactive Practice

Matrices Are Vectors Too

IKey Formulas

Concept	Formula	For $n = 3$
All $n \times n$ matrices $M$	$\dim = n^2$	$9$
Upper triangular matrices	$\dim = \frac{n(n+1)}{2}$	$6$
Symmetric matrices	$\dim = \frac{n(n+1)}{2}$	$6$
Diagonal matrices	$\dim = n$	$3$

Key relationship: Upper Triangular $\cap$ Symmetric $=$ Diagonal

IIVisualization 1 — Where a Matrix Lives

Zero the below-diagonal entries and it's upper triangular; mirror them and it's symmetric; do both and it's diagonal.

Above-diagonal Below-diagonal

M (3×3)

IIIVisualization 2 — One Vector, Two Costumes

The diagonal matrix $aE_1 + bE_2 + cE_3$ and the column vector $(a, b, c)$ carry the exact same three numbers.

a 5 b −3 c 4

aE₁ + bE₂ + cE₃ (a matrix)

coordinate vector in R³

💡 Same three coordinates, different packaging: the diagonal $3 \times 3$ matrices and $\mathbb{R}^3$ are both $3$ -dimensional and behave identically under $+$ and scaling — a vector space isomorphism.

IVVisualization 3 — Free Entries Are the Dimension

A subspace's dimension is just its count of independently choosable entries — watch each count scale with $n$ .

Matrix size n 3

💡 ★ = free entry, = fixed by symmetry, 0 = forced zero. Upper triangular and symmetric always share the dimension $\tfrac{n(n+1)}{2}$ , yet they are different subspaces.

VQuiz Questions

Question 1

What is the dimension of the vector space of all $4 \times 4$ matrices?

✅ Correct! A 4×4 matrix has 16 independent entries, so the dimension is 16.

❌ Not quite. Count the total number of entries in a 4×4 grid — each one is a free parameter.

Show solution

Solution:

The space of all $n \times n$ matrices has dimension $n^2$ , because each of the $n^2$ entries is an independent free parameter.

For $n = 4$ : $\dim = 4^2 = \mathbf{16}$ .

A basis consists of the 16 matrices that each have a single 1 in one position and 0s everywhere else.

Question 2

True or False: For a set of matrices to form a vector space, it must be closed under matrix multiplication.

✅ Correct! Only addition and scalar multiplication matter — matrix multiplication is irrelevant to the vector space structure.

❌ Not quite. Review which operations actually appear in the vector space axioms.

Show solution

Solution: False.

A vector space requires only:

Closure under addition
Closure under scalar multiplication
A zero element (additive identity) and additive inverses

Matrix multiplication plays no role in the vector space axioms. This is a key insight from the video: vector spaces are defined entirely through addition and scalar multiplication.

Question 3

A basis for the subspace of $3 \times 3$ symmetric matrices contains how many elements?

✅ Correct! Three diagonal plus three above-diagonal entries give 6 free parameters.

❌ Not quite. In a symmetric matrix, below-diagonal entries mirror those above. Count only the truly independent entries.

Show solution

Solution:

A $3 \times 3$ symmetric matrix satisfies $a_{ij} = a_{ji}$ , so entries below the diagonal are completely determined by entries above.

Free entries:

3 diagonal entries: positions $(1,1), (2,2), (3,3)$
3 above-diagonal entries: positions $(1,2), (1,3), (2,3)$

Total free entries $= 3 + 3 = 6 = \frac{3(3+1)}{2}$

The dimension is 6, so a basis has 6 elements.

Question 4

Which of the following is NOT a subspace of the space of all $3 \times 3$ matrices?

✅ Correct! The zero matrix is not invertible, so the set of invertible matrices fails the most basic subspace requirement.

❌ Not quite. Think about which set fails the most basic requirement: containing the zero matrix.

Show solution

Solution: Invertible matrices.

A subspace must contain the zero vector (the zero matrix). The zero matrix has determinant 0, so it is not invertible. Therefore invertible matrices cannot form a subspace.

Additionally, invertible matrices are not closed under addition: $I + (-I) = O$ (the zero matrix), which is not invertible.

The other three all pass the subspace test:

Upper triangular: closed under $+$ and $c\,\cdot$ , contains $O$ ✓
Symmetric: closed under $+$ and $c\,\cdot$ , contains $O$ ✓
Diagonal: closed under $+$ and $c\,\cdot$ , contains $O$ ✓

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