# Dimension and rank

### Dimension and rank

#### Lessons

Dimension of a Subspace
The dimension of a non-zero subspace $S$ (usually denoted as dim $S$), is the # of vectors in any basis for $S$. Since the null space and column space is a subspace, we can find their dimensions.

Note: Dimension of the column space = rank

Finding the Rank of a matrix:
1. Find the basis for the column space
2. Count the # of vectors in the basis. That is the rank.
Shortcut: Count the # of pivots in the matrix

Finding the dimensions of the null space:
1. Find the basis for the null space
2. Count the # of vectors in the basis. That is the dimension.
Shortcut: Count the # of free variables in the matrix.

The Rank Theorem
If a matrix $A$ has $n$ columns, then rank $A+$ dim $N(A) = n$.
• Introduction
Dimension and Rank Overview:
a)
Dimension of a subspace
• Dimension = number of vectors in the basis
• Can we find dimension of column space and null space?

b)
Rank of a Matrix
• Find the basis
• Count the # of vectors
• Shortcut = count the # of pivots

c)
Dimension of the Null Space
• Find the general solution
• Put in parametric vector form
• Count the # of vectors
• Shortcut = count the # of free variables

d)
The Rank Theorem
• Rank $A +$ dim $N(A) = n$
• An example of using the theorem

• 1.
Finding the Rank of a matrix
Find the rank of $A$ if: • 2.
Find a basis for the subspace spanned by the given vectors. What is the dimension of the subspace? • 3.
Finding the dimension of the null space
Find the dimension of the null space of $A$ if: • 4.
Utilizing the Rank Theorem
You are given the matrix $A$ and the echelon form of $A$. Find the basis for the column space, and find the rank and the dimensions of the null space. • 5.
Understanding the Theorems
Let the subspace of all solutions of $Ax=0$ have a basis consisting of four vectors, where $A$ is $4 \times 6$. What is the rank of $A$?

• 6.
Let $A$ be a $m \times n$ matrix where the rank of $A$ is $p$. Then what is the dimension of the null space of $A$?

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6.
Subspace of $\Bbb{R}^n$
6.1
Properties of subspace
6.2
Column space
6.3
Null space
6.4
Dimension and rank