Question

We often write vectors in \(\mathbb{R}^{n}\) as rows. If \(A\) is an \(m \times n\) matrix, show that, for each invertible \(n \times n\) matrix \(V, \operatorname{im}(A)=\operatorname{im}(A V)\)

Short answer

The image of matrix \(A\) is the same as the image of \(AV\) when \(V\) is invertible.

Step-by-step solution

Understanding the Image of a Matrix

The image (or column space) of a matrix \(A\), denoted \(\operatorname{im}(A)\), is the set of all possible linear combinations of its columns. This represents the span of the column vectors of \(A\). Our task is to show that this span is the same whether we consider \(A\) or \(AV\), when \(V\) is invertible.

Expressing the Image of AV

Consider the matrix product \(AV\), where \(V\) is an invertible \(n \times n\) matrix. The columns of \(AV\) are linear combinations of the columns of \(A\). Therefore, any column of \(AV\) can be expressed as \(A\cdot v_j\), where \(v_j\) is the \(j\)-th column of \(V\). Thus, the image of \(AV\) is spanned by these combinations.

Using the Invertibility of V

Since \(V\) is invertible, there exists a matrix \(V^{-1}\) such that \(VV^{-1} = I_n\), the \(n\times n\) identity matrix. Multiplying both sides of the equation \(AVz = b\) by \(V^{-1}\) gives \(A(Vz) = b\). The invertibility ensures that the transformation by \(V\) is reversible, allowing us to translate between vectors in the image of \(AV\) and \(A\).

Showing \(\operatorname{im}(A) = \operatorname{im}(AV)\)

For \(y \in \operatorname{im}(AV)\), there exists \(x\) such that \((AV)x = y\). Thus, \(A(Vx) = y\), which confirms \(y \in \operatorname{im}(A)\). Conversely, for any \(z \in \operatorname{im}(A)\), \(z = Aw\) for some \(w\). Since \(V\) is invertible, select \(w = Vx\), hence \(Az = AVx = z\). Therefore, any \(z\) in \(\operatorname{im}(A)\) is expressible as a combination from \(\operatorname{im}(AV)\).

Conclusion of the Proof

Both directions of inclusion are established: every vector in \(\operatorname{im}(AV)\) is in \(\operatorname{im}(A)\), and every vector in \(\operatorname{im}(A)\) can be formed from linear combinations in \(\operatorname{im}(AV)\). Hence, \(\operatorname{im}(A) = \operatorname{im}(AV)\).

Key concepts

Matrix Invertibility

Matrix invertibility is a fundamental concept in linear algebra. It tells us whether a square matrix has a unique inverse, such that when multiplied with the original gives the identity matrix. For a matrix to be invertible, also known as non-singular, it must fulfill certain conditions:

• The matrix must be square, meaning it has the same number of rows and columns.
• The determinant of the matrix must not be zero.

The identity matrix is crucial in this context as it acts as the multiplicative identity in matrix multiplication. If you have an invertible matrix, say matrix \( V \), there exists a matrix \( V^{-1} \) so that \( VV^{-1} = V^{-1}V = I_n \), where \( I_n \) is the identity matrix of size \( n \). This property is essential for transforming and simplifying systems of equations or matrix equations.
Matrix invertibility is key in various applications, such as solving linear systems, and in the context of linear transformations, ensures we can "undo" or reverse the transformation.

Column Space

The column space of a matrix, also known as the image or range, is another important topic in linear algebra. It represents the set of all linear combinations of the matrix's column vectors. Essentially, it's the span formed by the columns of the matrix.

Consider a matrix \( A \) with columns \( a_1, a_2, ..., a_n \). The column space of \( A \), denoted \( \ ext{im}(A) \), is the span of these column vectors:

\[ \ ext{im}(A) = \ ext{Span}(a_1, a_2, ..., a_n) \]

The column space plays a crucial role in understanding the solutions to a system of linear equations represented by \( Ax = b \). If \( b \) is in the column space of \( A \), there exists a solution to the equation. The concept of column space is also crucial when discussing concepts like rank, dimension, and basis of a vector space, which provide valuable information about the structure of the matrix and the linear transformation it represents.

Matrix Multiplication

Matrix multiplication is an operation that takes two matrices and produces another matrix. It's not as straightforward as multiplying numbers, as it must adhere to specific rules.

When multiplying two matrices, \( A \, (m \times n) \) and \( B \, (n \times p) \), the resulting matrix \( C \) is of size \( (m \times p) \). Each element \( c_{ij} \) in the result is calculated as the dot product of the \( i \)-th row of matrix \( A \) and the \( j \)-th column of matrix \( B \).

It's important to remember:

• The order of multiplication matters; \( AB \) is generally not the same as \( BA \).
• Matrix multiplication is associative; that is, \( (AB)C = A(BC) \).
• Matrix multiplication is distributive; that is, \( A(B + C) = AB + AC \).

This operation is significant for various processes, including linear transformations, where one matrix can represent a transformation, and another can represent its modification or sequence.

Vector Spaces

In linear algebra, a vector space is a collection of vectors where two operations are defined: vector addition and scalar multiplication. These operations should satisfy particular axioms or properties, like distributive, associative, and commutative properties, to name a few.

Consider a set \( V \), a vector space when:

• It contains the zero vector.
• It is closed under vector addition; that is, adding any two vectors in \( V \) must also be in \( V \).
• It is closed under scalar multiplication, meaning multiplying a vector in \( V \) by a scalar still results in a vector within \( V \).

Vector spaces are significant because they provide a framework where vectors operate similarly to the familiar number system. This framework allows for the abstraction and application of many linear algebra concepts across different dimensions and fields, including spaces of functions, polynomials, and number sequences.