# Where are these subspaces?

\mathcal{C}(A)
\mathcal{N}(A^\top)
inside~\mathbb{R}^m
\mathcal{C}(A^\top)
\mathcal{N}(A)
inside~\mathbb{R}^n

# What are their dimensions?

\mathcal{C}(A)
\mathcal{N}(A^\top)
inside~\mathbb{R}^m
\mathcal{C}(A^\top)
\mathcal{N}(A)
inside~\mathbb{R}^n
dim=r
dim=n-r
dim=r
dim=m-r

# What are their basis vectors?

### Basis for $$\mathcal{C}(A)$$

\begin{bmatrix} 1&1&1&1\\ 0&1&2&3\\ 0&0&0&0 \end{bmatrix}
\begin{bmatrix} 1&0&-1&-2\\ 0&1&2&3\\ 0&0&0&0 \end{bmatrix}
\begin{bmatrix} 1&1&1&1\\ 1&2&3&4\\ 2&3&4&5 \end{bmatrix}
A
U
R

### Special solutions for $$A\mathbf{x} = \mathbf{0}$$

\begin{bmatrix} 1&2\\ -2&-3\\ 1&0\\ 0&1 \end{bmatrix}
X

# What are their basis vectors?

### Basis for $$\mathcal{N}(A^\top)$$

\begin{bmatrix} 1&1&1&1\\ 0&1&2&3\\ 0&0&0&0 \end{bmatrix}
\begin{bmatrix} 1&0&-1&-2\\ 0&1&2&3\\ 0&0&0&0 \end{bmatrix}
\begin{bmatrix} 1&1&1&1\\ 1&2&3&4\\ 2&3&4&5 \end{bmatrix}
A
U
R

### The rows of $$E$$ corresponding to the 0 rows of $$R$$

\begin{bmatrix} A & I \end{bmatrix}
E

### How did we get $$R$$ ?

= \begin{bmatrix} R & E \end{bmatrix}
r2 = r2 - r1
r3 = r3 - 2r1
r3 = r3 - r2
\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}
\begin{bmatrix} 1&0&0\\ -1&1&0\\ -1&-1&1 \end{bmatrix}
r1 = r1 - r2
\begin{bmatrix} 2&-1&0\\ -1&1&0\\ -1&-1&1 \end{bmatrix}

E

# What are their basis vectors?

A
U
R

### The rows of $$E$$ corresponding to the 0 rows of $$R$$

\begin{bmatrix} A & I \end{bmatrix}
E

### How did we get $$R$$ ?

= \begin{bmatrix} R & E \end{bmatrix}
r2 = r2 - r1
r3 = r3 - r1
\begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix}
r3 = r3 - 2r2

### These linear combinations of the rows of A produce the 0 vector (hence they are solutions to $$A^\top\mathbf{x} = \mathbf{0}$$ )

E
\begin{bmatrix} 1&1&2\\ 0&1&1\\ 0&0&0\\ 0&0&0\\ \end{bmatrix}
\begin{bmatrix} 1&0&1\\ 0&1&1\\ 0&0&0\\ 0&0&0\\ \end{bmatrix}
\begin{bmatrix} 1&1&2\\ 1&2&3\\ 1&3&4\\ 1&4&5\\ \end{bmatrix}
r4 = r4 - r1
r4 = r4 - 3r2
r1 = r1 - r2
\begin{bmatrix} 1&0&0&0\\ -1&1&0&0\\ 1&-2&1&0\\ 2&-3&0&1 \end{bmatrix}
\begin{bmatrix} 2&-1&0&0\\ -1&1&0&0\\ 1&-2&1&0\\ 2&-3&0&1 \end{bmatrix}

# Practice Problems

### For each of the above matrices, find the dimensions and the basis of the 4 fundamental subspaces

\begin{bmatrix} 1&2&3\\ 1&1&2\\ 2&1&0 \end{bmatrix}
\begin{bmatrix} 1&2&3&3&2&1\\ 1&2&2&1&1&2\\ 2&0&0&1&2&2 \end{bmatrix}
\begin{bmatrix} 0&2&2\\ 1&2&1\\ 2&1&3\\ 3&1&1\\ -1&1&-2\\ 2&1&0\\ \end{bmatrix}
\begin{bmatrix} 1&2&3&3&2&1\\ 2&1&0&1&2&2\\ 4&5&6&7&6&4 \end{bmatrix}

# $$\mathcal{N}(A^\top)$$ is left null space of $$A$$

A^\top \mathbf{y} = 0
Let~\mathbf{y}\in \mathcal{N}(A^\top)

### Taking transpose on both sides

\mathbf{y}^\top A = 0^\top