Question

If \(A\) is \(m \times n\) and \(\mathbf{b}\) is \(m \times 1,\) show that \(\mathbf{b}\) lies in the column space of \(A\) if and only if \(\operatorname{rank}[A \mathbf{b}]=\operatorname{rank} A\)

Short answer

\(\mathbf{b}\) lies in the column space of \(A\) if and only if \(\operatorname{rank}[A \mathbf{b}] = \operatorname{rank} A\).

Step-by-step solution

Understand Column Space

The column space of a matrix \(A\) is the set of all possible linear combinations of its column vectors. A vector \(\mathbf{b}\) lies in the column space of \(A\) if it can be expressed as \(A\mathbf{x}\) for some vector \(\mathbf{x}\).

Define Augmented Matrix

Consider the augmented matrix \([A \mathbf{b}]\), which is formed by appending \(\mathbf{b}\) as an additional column to \(A\).

Recall Rank Definition

The rank of a matrix is the dimension of its column space or row space, which is equivalent to the number of pivot columns in its row echelon form.

Equivalent Rank Condition for Column Space

For \(\mathbf{b}\) to be in the column space of \(A\), \(\mathbf{b}\) must be a linear combination of the columns of \(A\). This is true when the rank of the matrix \([A \mathbf{b}]\) is equal to the rank of \(A\).

Forward Implication (\(\Rightarrow\))

Assume \(\mathbf{b}\) is in the column space of \(A\). There must exist a solution \(\mathbf{x}\) such that \(A\mathbf{x} = \mathbf{b}\). Thus, including \(\mathbf{b}\) as a column in \(A\) does not increase the number of pivot columns, implying \(\operatorname{rank}[A \mathbf{b}] = \operatorname{rank} A\).

Reverse Implication (\(\Leftarrow\))

Assume \(\operatorname{rank}[A \mathbf{b}] = \operatorname{rank} A\). This means adding \(\mathbf{b}\) as a column does not introduce any new pivot columns beyond those already in \(A\). Therefore, \(\mathbf{b}\) must be expressible as a linear combination of the columns of \(A\), implying \(\mathbf{b}\) is in the column space of \(A\).

Conclusion

Thus, \(\mathbf{b}\) lies in the column space of \(A\) if and only if \(\operatorname{rank}[A \mathbf{b}] = \operatorname{rank} A\).

Key concepts

Matrix Rank

Understanding the rank of a matrix is crucial in linear algebra as it provides insight into the solutions of linear equations associated with the matrix. The rank of a matrix is defined as the dimension of its column space or row space. Another way to view this is by considering the number of pivot columns the matrix has when it is in its row echelon form. A higher rank indicates more linearly independent columns, showing that the matrix can span a larger dimensional space.
- Matrix rank helps determine if the matrix's columns are linearly independent. - If some columns depend on others, the rank will be less than the total number of columns. - Understanding rank is vital for solving systems of linear equations, as it can tell us how many solutions exist.

Linear Combination

A linear combination is a fundamental concept in understanding how vectors relate to each other in a vector space. To form a linear combination of a set of vectors, multiply each vector by a scalar (a real number) and sum the results. This process is essential in determining whether a vector lies in a specific space, such as the column space of a matrix.

• The column space of a matrix consists of all linear combinations of its column vectors.
• If a vector can be expressed as a linear combination of the matrix’s columns, it lies in the column space.
• This idea helps in understanding whether a vector can be obtained through a given transformation or system represented by the matrix.

Linear combinations are pivotal in various applications, such as solving linear systems or finding bases for spaces.

Augmented Matrix

An augmented matrix is used to solve systems of linear equations and is particularly helpful when appending vectors to analyze their relationship with a given matrix. When you add a vector \(\mathbf{b}\) to a matrix \(A\) to form an augmented matrix \[ [A \mathbf{b}] \], you're trying to see if \(\mathbf{b}\) lies in the column space of \(A\). - The augmented matrix provides a systematic way to explore the relationship between \(\mathbf{b}\) and the matrix \(A\).- By analyzing the rank of this augmented matrix, one can determine if \(\mathbf{b}\) is a linear combination of \(A\)'s columns.- It combines the techniques of row operations and rank to study the intersection of a vector with a vector space.Understanding augmented matrices is essential in exploring vector spaces and solving matrix equations.

Pivot Columns

Pivot columns are crucial elements in understanding the structure of a matrix. They are the columns in a matrix that contain leading ones when the matrix is transformed into row echelon form. These columns indicate the presence of linearly independent columns in the matrix. - Identifying pivot columns allows us to determine the rank of the matrix. - The row reduction process identifies these columns and simplifies matrix-related problems, like determining independence and dimension. - Pivot columns help understand whether an added vector (through an augmented matrix) introduces a new direction in the space. Determining pivot columns is a foundational skill for navigating through the properties of matrices and systems of linear equations.