Find a redundant column vector of the given matrix A, and...

Find a redundant column vector of the given matrix $A$, and write it as a linear combination of preceding columns. Use this representation to write a nontrivial relation among the columns, and thus find a nonzero vector in the kernel of A. (This procedure is illustrated in Example $8 .$) $$\left[\begin{array}{ll} 0 & 1 \\ 0 & 2 \end{array}\right]$$

Find a redundant column vector of the given matrix $A$, and write it as a linear combination of preceding columns. Use this representation to write a nontrivial relation among the columns, and thus find a nonzero vector in the kernel of A. (This procedure is illustrated in Example $8 .$)
$$\left[\begin{array}{ll}
0 & 1 \\
0 & 2
\end{array}\right]$$

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Key Concepts

Kernel (Null Space) of a Matrix

The kernel, or null space, of a matrix is the set of all vectors that, when multiplied by the matrix, yield the zero vector. Finding a nonzero vector in the kernel, often by exploiting linear dependencies among the columns, is a key outcome when one column is redundant.

Linear Dependence

Linear dependence is a relationship between vectors where at least one vector can be written as a linear combination of the others. When such a relationship exists among the columns of a matrix, it implies redundancy and is essential for identifying nontrivial solutions in the kernel of the matrix.

Redundant Column

A redundant column in a matrix is one that can be expressed as a linear combination of other columns in the matrix. This property indicates that the column does not add any new dimensions to the column space of the matrix, meaning it is linearly dependent on the other columns.

Linear Combination

A linear combination involves forming a new vector by multiplying existing vectors by scalar coefficients and then summing the results. In the context of matrices, representing one column as a linear combination of preceding columns is a way to express its dependency on those columns.

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