Subspaces, dimension, and the kernel

Span, linear independence and basis, the rank-nullity relation, and describing every solution of a homogeneous system as the null space of its matrix.

Notes on Diffy Qs: Differential Equations for Engineers, Subspaces, dimension, and the kernel

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3 exercises

Exercise 101

For the following sets of vectors, find a basis for the subspace spanned by the vectors, and find the dimension of the subspace.

$\begin{bmatrix} 1 \\ 2 \end{bmatrix} , \quad \begin{bmatrix} 1 \\ 1 \end{bmatrix}$
$\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} , \quad \begin{bmatrix} 2 \\ 2 \\ 2 \end{bmatrix} , \quad \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}$
$\begin{bmatrix} 5 \\ 3 \\ 1 \end{bmatrix} , \quad \begin{bmatrix} 5 \\ -1 \\ 5 \end{bmatrix} , \quad \begin{bmatrix} -1 \\ 3 \\ -4 \end{bmatrix}$
$\begin{bmatrix} 2 \\ 2 \\ 4 \end{bmatrix} , \quad \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} , \quad \begin{bmatrix} 4 \\ 4 \\ -3 \end{bmatrix}$
$\begin{bmatrix} 1 \\ 0 \end{bmatrix} , \quad \begin{bmatrix} 2 \\ 0 \end{bmatrix} , \quad \begin{bmatrix} 3 \\ 0 \end{bmatrix}$
$\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \quad \begin{bmatrix} 2 \\ 0 \\ 0 \end{bmatrix} , \quad \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}$

2.5

Exercise 102

For the following matrices, find a basis for the kernel (nullspace).

$\begin{bmatrix} 2 & 6 & 1 & 9 \\ 1 & 3 & 2 & 9 \\ 3 & 9 & 0 & 9 \end{bmatrix}$
$\begin{bmatrix} 2 & -2 & -5 \\ -1 & 1 & 5 \\ -5 & 5 & -3 \end{bmatrix}$
$\begin{bmatrix} 1 & -5 & -4 \\ 2 & 3 & 5 \\ -3 & 5 & 2 \end{bmatrix}$
$\begin{bmatrix} 0 & 4 & 4 \\ 0 & 1 & 1 \\ 0 & 5 & 5 \end{bmatrix}$

2.5

Exercise 103

Suppose the column space of a $9 \times 5$ matrix $A$ is of dimension 3. Find

Rank of $A$ .
Nullity of $A$ .
Dimension of the row space of $A$ .
Dimension of the nullspace of $A$ .
Size of the maximum subset of linearly independent rows of $A$ .

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