Let T be multiplication by the matrix A . Find (a) a basis...

Let $T$ be multiplication by the matrix $A .$ Find
(a) a basis for the range of $T$
(b) a basis for the kernel of $T$
(c) the rank and nullity of $T$.
(d) the rank and nullity of $A$.
$A=\left[\begin{array}{rrr}2 & 0 & -1 \\ 4 & 0 & -2 \\ 20 & 0 & 0\end{array}\right]$

Key Concepts

Linear Transformation

A linear transformation is a mapping between vector spaces that preserves the operations of addition and scalar multiplication. When defined by multiplying a vector by a matrix, the properties of the transformation (such as its range and kernel) are directly related to the structure of the matrix.

Range (Column Space)

The range, or column space, of a linear transformation is the set of all possible outputs of the transformation. It is spanned by the columns of the matrix, and finding a basis for this space involves identifying the linearly independent columns.

Kernel (Null Space)

The kernel, or null space, of a linear transformation is the set of all vectors that are mapped to the zero vector by the transformation. A basis for the kernel is found by solving the homogeneous equation Ax = 0, which reveals the dependencies among the vector components.

Rank

The rank of a linear transformation or matrix is the dimension of its range (column space). It indicates the number of linearly independent columns in the matrix, showing how many dimensions of the output space are covered by the transformation.

Nullity

The nullity of a linear transformation or matrix is the dimension of its kernel (null space). It represents the number of free parameters or independent solutions in the homogeneous equation Ax = 0, and relates to the overall dimensionality via the Rank-Nullity Theorem.

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