Question

describe the column space (range ) and the nullspace (kernel) of the matrices
A= 1 -1 B= 0 0 0
0 0 0 0 0

Answer 1

Conditions

describe the column space (range) and the nullspace (kernel) of the matrices

A= 1 -1 B=0 0 0

0 0 0 0 0

Solution

In linear algebra, the column space, $C(A)$ of a matrix (sometimes called the range of a matrix) is the set of all possible linear combinations of its column vectors. The column space of an $m \times n$ matrix is a subspace of $m$-dimensional Euclidean space. The dimension of the column space is called the rank of the matrix. The column space of a matrix is the image or range of the corresponding matrix transformation.

$$A = \left( \begin{array}{cc} 1 & -1 \\ 0 & 0 \end{array} \right)$$

Let's consider a linear combination for A:

$$c_1 v_1 + c_2 v_2$$

where $c_1, c_2$ are scalars. The set of all possible linear combinations of $v_1, v_2$ is called the column space of A:

$$c_1 \left( \begin{array}{c} 1 \\ 0 \end{array} \right) + c_2 \left( \begin{array}{c} -1 \\ 0 \end{array} \right) = \left( \begin{array}{c} c_1 \\ 0 \end{array} \right) + \left( \begin{array}{c} -c_2 \\ 0 \end{array} \right) = \left( \begin{array}{c} c_1 - c_2 \\ 0 \end{array} \right)$$

In this case, the column space is precisely the set of vectors $(x, 0) \in R^2$ for all $x \in R$.

B:

$$B = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right)$$

It's a zero matrix.

Let's consider a linear combination for B:

$$c_1 \left( \begin{array}{c} 0 \\ 0 \end{array} \right) + c_2 \left( \begin{array}{c} 0 \\ 0 \end{array} \right) + c_3 \left( \begin{array}{c} 0 \\ 0 \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \end{array} \right)$$

In this case, the column space is null vector $\left( \begin{array}{c} 0 \\ 0 \end{array} \right)$.

In linear algebra, the kernel or null space (also nullspace) of a matrix $A$ is the set of all vectors $x$ for which $Ax = 0$. The kernel of a matrix with $n$ columns is a linear subspace of $n$-dimensional Euclidean space. The dimension of the null space of $A$ is called the nullity of $A$.

A:

$$Ax = \begin{pmatrix} 1 & -1 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} x_1 - x_2 \\ 0 \end{pmatrix} = 0$$ $$x_1 - x_2 = 0$$ $$x_1 = x_2$$

The kernel of matrix A are all vectors $(x_1, x_2) \in R^2$, where $x_1 = x_2, x_2 \in R$.

B:

$$Bx = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$$

This means that the kernel of matrix B are all vectors $(x_1, x_2) \in R^2$, or $R^2$ itself.

Answer

C(A): $(x, 0) \in R^2$ for all $x \in R$

C(B) is null vector $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$

Ker(A): $(x_1, x_2) \in R^2$, where $x_1 = x_2, x_2 \in R$

Ker(B): $R^2$