Lecture notes · Linear Algebra · Lecture 9

Independence, basis, and dimension

MIT 18.06 Linear Algebra, lecture 9, explained in plain English

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Part 1 · 0:00–10:00

This part explains when a group of vectors counts as independent or dependent, and why a system with more unknowns than equations always has a non-zero solution.

Key ideas

The lecture says we talk about a bunch of vectors being independent, not a matrix being independent. Words like basis and dimension will get exact meanings later in the lecture. 0:58
If a matrix has more unknowns than equations, then Ax = 0 has some solution that is not all zeros. Elimination always leaves at least one free variable, and you can give that free variable any non-zero value. 2:15
Vectors are independent when no mix of them adds up to the zero vector, except the mix where every amount is zero. If some other mix gives zero, they are dependent. 6:06
Two vectors where one is twice the other are dependent. Two copies of the first, minus the second, gives the zero vector. 8:17
If any vector in the group is the zero vector, the group is dependent. You can use that zero vector alone, with any amount you like, and still get zero. 9:31

Words to know

linearly independent: A group of vectors where no mix of them lands on zero, unless every amount in the mix is zero.
dependent: The opposite of independent. Some mix of the vectors, with at least one amount not zero, adds up to the zero vector.
combination: Taking some amount of each vector and adding them together, like a recipe that uses a set amount of each ingredient.
free variable: A variable that has no pivot in the row-reduced system, so you can choose its value yourself.

Example

Think of walking directions. If one step says go 2 blocks north and another says go 4 blocks north, the second step adds nothing new. You can mix the two steps, going forward and backward, and end up exactly where you started. That is what dependence looks like.

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Part 2 · 10:00–20:00

This part explains when a set of vectors is independent or dependent, and what it means for vectors to span a space.

Key ideas

Vectors are dependent if you can mix them with some non-zero amounts and get the zero vector. They are independent if the only way to get zero is to use zero of every vector. 10:19
You can test independence by putting the vectors into the columns of a matrix. The columns are independent exactly when the null space of that matrix contains only the zero vector. 14:10
If the columns are independent, every column is a pivot column, so the rank equals the number of columns. If the rank is smaller than the number of columns, the columns are dependent. 16:11
To span a space means that all combinations of the vectors fill up that space. The columns of a matrix span its column space in exactly this way. 19:24

Words to know

independent: A set of vectors is independent if no vector in the set can be built by mixing the others. The only way to get the zero vector from them is to use zero of each.
null space: The collection of all input vectors that a matrix turns into the zero vector. It is a set of answers to the question: which mix of the columns gives zero?
span: To span a space means the vectors can be mixed in every possible way to reach every point in that space.
rank: A count of how many columns in a matrix are pivot columns. It tells you how many independent directions the columns cover.

Example

Think of mixing paint colors. If you have red and blue, you can make many purples, but if you already have a purple made from red and blue, adding a third color that you can copy from the first two adds nothing new. That is like dependent vectors: one of them is already made from the others.

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Part 3 · 20:00–30:00

This part explains what a basis is: a small set of vectors that both spans a space and is independent, and it works through examples in three-dimensional space.

Key ideas

The span of a set of vectors is every combination of them, and it is the smallest space that contains those vectors. 20:00
A basis is a sequence of vectors with two properties: they are independent, and they span the space. Independent means no vector is needed as a mix of the others, and span means they reach the whole space. 22:36
The standard x, y, z vectors form a basis for three-dimensional space. You can check independence by asking whether only all-zero coefficients make the combination equal the zero vector. 25:18
Two vectors can be independent and still fail to be a basis for R^3, because they do not span the space. A third vector must lie outside the plane of the first two, so 3 3 7 would fail and 3 3 8 would work. 27:31
For R^n, n vectors form a basis when they fit as the columns of a square matrix that works out correctly. The lesson stops before giving the final test, so this part is not fully finished. 29:22

Words to know

span: All the vectors you can make by mixing the given vectors together with scaling and adding. The span is the smallest space that contains them.
independent: A set of vectors is independent when none of them can be made by mixing the others. The only way to mix them into the zero vector is to use zero for every amount.
basis: A small, exact list of vectors that is independent and also spans the whole space. It tells you everything about that space.
null space: The set of all vectors that a matrix turns into the zero vector. If only the zero vector works, the matrix's columns are independent.

Example

Think of a set of building blocks like LEGO bricks in three basic sizes. If the blocks cannot be built from each other, and you can build any shape with them, they act like a basis. A third brick that is just a stack of the first two adds nothing new, so it is a wasted piece.

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Part 4 · 30:00–40:00

This part explains that a basis has a fixed size for a given space, and that size is called the dimension.

Key ideas

A set of vectors can be a basis for a space only if the vectors are independent and span that space. For a square matrix in R^n, the columns form a basis exactly when the matrix is invertible. 30:00
A basis is not unique. Many different sets of vectors can be bases for the same space, such as the columns of any invertible 3 by 3 matrix for R^3. 32:56
Every basis for a given space has the same number of vectors. If one basis has six vectors, every basis for that space has six. 35:47
That shared number of vectors is called the dimension of the space. It tells you how big the space is. 36:29
A set of vectors can span a space without being a basis. In the example, four column vectors span the column space, but they are not independent, so they are not a basis. 39:33

Words to know

basis: A small set of vectors that is independent and also spans the whole space. Every vector in the space can be built from it.
independent: No vector in the set can be made from the others. The only way to combine them to get zero is to use zero of each.
span: All the combinations you can make from a set of vectors. The span is the space those combinations fill.
dimension: The number of vectors in any basis for a space. It measures how big the space is.

Example

Think of a set of LEGO bricks that can build any shape in a box. A good set has just enough bricks, with no spare copies that can be made from the others. You can pick different brick sets, but each one needs the same number of bricks for that box. That number is like the dimension.

3 checked practice questions for this part are in the app.

Part 5 · 40:00–49:58

This part explains how to find a basis (a minimal set of building-block vectors) for the column space of a matrix, how the rank gives its dimension, and how the null space's dimension is counted from free variables.

Key ideas

The pivot columns of a matrix form a basis for its column space. Go left to right and keep each column that is not a combination of the ones you already kept. 40:17
The rank of a matrix is the number of pivot columns, and that number is also the dimension of the column space. Rank describes a matrix, while dimension describes a space. 41:23
A basis is not unique. Many different sets of vectors can work, so the lecturer builds a second, different basis to show this. 43:16
If you already know the dimension of a space, and you have that many vectors that are independent, they automatically span the space and form a basis. 45:01
Special solutions for the null space come from setting one free variable to 1 and the others to 0. The number of free variables is n minus r, which gives the dimension of the null space. 47:20

Words to know

Basis: A small set of vectors that can build every vector in a space by mixing them, with none of them redundant.
Column space: All the vectors you can make by mixing the columns of a matrix together.
Rank: The number of pivot columns in a matrix, which is also the dimension of its column space.
Null space: All the input vectors that a matrix turns into zero.

Example

Think of a box of paint colors where some colors can be mixed from others. The basis is the smallest set of paints you need to make every color in the box, and the dimension is how many paints that set has. Your rank is the count of paints that cannot be made from the others.

3 checked practice questions for this part are in the app.

← Lecture 3: Multiplication and inverse matrices · Lecture 21: Eigenvalues and eigenvectors →

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. MIT OpenCourseWare · CC BY-NC-SA 4.0 · not affiliated with or endorsed by MIT. These notes are shared under the same license.

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