Find the redundant column vectors of the given matrix A "by...

Find the redundant column vectors of the given matrix $A$ "by inspection." Then find $a$ basis of the image of A and a basis of the kernel of $A$.
$$\left[\begin{array}{rrr}
1 & -2 & 3 \\
2 & 4 & 6
\end{array}\right]$$

Key Concepts

Redundant Column Vectors

This concept refers to identifying columns in a matrix that can be expressed as a linear combination of other columns, meaning they do not contribute any new information to the span of the column space. Recognizing redundant vectors is important for simplifying the matrix and understanding its structure, particularly for determining the rank of the matrix.

Linear Independence and Dependency

A set of vectors is said to be linearly independent if no vector in the set can be written as a linear combination of the others. Conversely, if at least one vector can be represented as such a combination, the set exhibits linear dependency. These concepts are foundational for determining a matrix’s rank and for selecting a minimal subset of vectors that span a given space.

Column Space (Image) of a Matrix

The image or column space of a matrix is the set of all linear combinations of its column vectors. It represents all the possible outputs of the corresponding linear transformation. Finding a basis for the image involves selecting a subset of the matrix's columns that are linearly independent and span this space, thereby giving insight into the range of the transformation.

Kernel (Null Space) of a Matrix

The kernel or null space of a matrix is the set of all vectors that the matrix maps to the zero vector. It represents the solutions to the homogeneous linear system associated with the matrix. Identifying a basis for the kernel is essential in understanding the structure of the solution space and the dependence relations among the columns.

Basis

A basis of a vector space is a set of linearly independent vectors that span the entire space. In the context of matrices, bases are determined for both the column space and the kernel. The basis provides the most efficient way to describe all elements within the space since every vector in the space can be uniquely represented as a linear combination of the basis vectors.

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