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 $A$.
$$\left[\begin{array}{lll}
1 & 2 & 1 \\
1 & 2 & 2 \\
1 & 2 & 3
\end{array}\right]$$

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Key Concepts

Linear Independence

Linear independence refers to a set of vectors where no vector can be expressed as a linear combination of the others. This concept is key when deciding which columns contribute uniquely to the span of a space, thereby identifying any redundant (dependent) columns in a matrix.

Column Space (Image)

The column space, or image, of a matrix is the set of all possible linear combinations of its column vectors. A basis for the column space consists of a minimal set of linearly independent columns, meaning that once redundancy is removed, these columns span the entire image of the matrix.

Kernel (Null Space)

The kernel, or null space, of a matrix is the set of all vectors that, when multiplied by the matrix, yield the zero vector. Determining a basis for the kernel involves solving the homogeneous system, and it reveals the dependencies and solutions where the transformation represented by the matrix collapses vectors to zero.

Redundant (Dependent) Columns

Redundant columns in a matrix are those that can be expressed as linear combinations of other columns; they do not add new dimensions to the column space. Recognizing these columns by inspection simplifies the process of identifying a basis for the image of the matrix and understanding its overall structure.

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