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 $8 .$) $$\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]$$

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 $8 .$)
$$\left[\begin{array}{ll}
1 & 1 \\
1 & 1
\end{array}\right]$$

Key Concepts

Kernel (Null Space) of a Matrix

The kernel or null space of a matrix consists of all vectors that, when multiplied by the matrix, result in the zero vector. It reveals the solutions to the homogeneous system associated with the matrix and reflects the inherent redundancies or degrees of freedom within the matrix. Finding a nontrivial vector in the kernel demonstrates the existence of linear dependencies among the columns of the matrix.

Linear Dependence

Linear dependence occurs when one vector in a set can be written as a linear combination of the others. If even one vector in a collection can be derived from the others, the set is linearly dependent. This concept is crucial in determining the basis of a space and in analyzing the rank and kernel of matrices.

Redundant Column Vector

A redundant column vector in a matrix is one that can be expressed as a linear combination of other columns in the matrix. This means that the vector does not contribute new, independent information—its presence does not expand the column space of the matrix. In the context of redundancy, identifying such a column is essential for simplifying systems and understanding the inherent dependencies among the vectors.

Linear Combination

A linear combination involves constructing a new vector by multiplying existing vectors by scalar coefficients and adding the results. This concept is fundamental in linear algebra, as it allows the representation of vectors in terms of a basis or other subsets of vectors, thereby revealing the dependencies and structure within the vector space.

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