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 $A$.
$$\left[\begin{array}{lll}
1 & 0 & 0 \\
1 & 0 & 0 \\
1 & 1 & 1
\end{array}\right]$$

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Key Concepts

Linear Independence

Linear independence is the property of a set of vectors such that no vector in the set can be written as a linear combination of the others. This concept underpins the formation of bases for both the image and the kernel, ensuring that the chosen vectors contribute uniquely to their respective spaces.

Redundant Column Vectors

Redundant column vectors are those that can be expressed as a linear combination of other columns in a matrix. Identifying redundancy helps simplify the column space because repeated or dependent vectors do not add new directions, which simplifies finding a basis for the image.

Basis of the Image

The image (or column space) of a matrix consists of all linear combinations of its column vectors. A basis for the image is a minimal set of linearly independent vectors that span this space. Identifying such a basis reveals the effective dimensions in which the matrix operates and is key to understanding its range.

Basis of the Kernel

The kernel (or null space) of a matrix includes all vectors that are mapped to the zero vector under the matrix transformation. A basis for the kernel is a collection of linearly independent vectors that generate this space. This concept is essential for understanding the solutions to homogeneous linear systems and the matrix's nullity.

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