LAFF From composing linear transformations to matrix-matrix multiplication

Subsection 4.4.2 From composing linear transformations to matrix-matrix multiplication

Homework 4.4.2.1.

Let \(L_A: \mathbb{R}^k \rightarrow \mathbb{R}^m \) and \(L_B: \mathbb{R}^n \rightarrow \mathbb{R}^k \) both be linear transformations and, for all \(x \in \mathbb{R}^n \text{,}\) define the function \(L_C: \mathbb{R}^n \rightarrow \mathbb{R}^m \) by \(L_C( x ) = L_A( L_B( x ) ) \text{.}\)

ALWAYS/SOMETIMES/NEVER: \(L_C(x) \) is a linear transformations.

Answer

ALWAYS

Now prove it!

Solution

Let \(x, y \in \mathbb{R}^n \) and \(\alpha \in \mathbb{R} \text{.}\)

\(L_C( \alpha x ) = L_A( L_B( \alpha x ) ) = L_A( \alpha L_B( x ) ) = \alpha L_A( L_B( x ) ) = \alpha L_C( x ) \text{.}\)
\(\begin{array}[t]{@{}rcl} L_C( x + y ) \amp=\amp L_A( L_B( x + y )) = L_A( L_B( x ) + L_B( y )) \\ \amp=\amp L_A( L_B( x )) + L_A( L_B( y )) = L_C( x ) + L_C( y ). \end{array}\)

This homework confirms that the composition of two linear transformations is itself a linear transformation.

Homework 4.4.2.2.

Let \(A \in \mathbb{R}^{m \times n} \text{.}\)

ALWAYS/SOMETIMES/NEVER: \(A^T A \) is well-defined. (By well-defined we mean that \(A^T A \) makes sense. In this particular case this means that the dimensions of \(A^T \) and \(A \) are such that \(A^T A \) can be computed.)

Answer

ALWAYS

Why?

Solution

\(A^T \) is \(n \times m \) and \(A \) is \(m \times n \text{,}\) and hence the column size of \(A^T \) matches the row size of \(A \text{.}\)

Homework 4.4.2.3.

Let \(A \in \mathbb{R}^{m \times n} \text{.}\)

ALWAYS/SOMETIMES/NEVER: \(A A^T \) is well-defined.

Answer

ALWAYS

Why?

Solution

Apply the result in the last exercise, with \(A \) replaced by \(A^T \text{.}\)

Now, let linear transformations \(L_A \text{,}\) \(L_B \text{,}\) and \(L_C \) be represented by matrices \(A \in \mathbb{R}^{m \times k} \text{,}\) \(B \in \mathbb{R}^{k \times n} \text{,}\) and \(C \in \mathbb{R}^{m \times n} \text{,}\) respectively. (You know such matrices exist since \(L_A \text{,}\) \(L_B \text{,}\) and \(L_C \) are linear transformations.) Then \(C x = L_C( x ) = L_A ( L_B ( x ) ) = A( B x ) \text{.}\)

Remark 4.4.2.1.

The matrix-matrix multiplication (product) is defined as the matrix \(C \) such that, for all vectors \(x \text{,}\) \(Cx = A( B ( x ) ) \text{.}\) The notation used to denote that matrix is \(C = A \times B \) or, equivalently, \(C = A B \text{.}\) The operation \(A B \) is called a matrix-matrix multiplication or product.

Remark 4.4.2.2.

If \(A \) is \(m_A \times n_A \) matrix, \(B \) is \(m_B \times n_B \) matrix, and \(C \) is \(m_C \times n_C \) matrix, then for \(C = A B \) to hold it must be the case that \(m_C = m_A \text{,}\) \(n_C = n_B \text{,}\) and \(n_A = m_B \text{.}\) Usually, the integers \(m \) and \(n \) are used for the sizes of \(C \text{:}\) \(C \in \mathbb{R}^{m \times n} \) and \(k \) is used for the ``other size'': \(A \in \mathbb{R}^{m \times k} \) and \(B \in \mathbb{R}^{k \times n} \text{:}\)