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In Exercises, view vectors in n as n × 1 matrices. For u
Chapter 2, Problem 27E(choose chapter or problem)
In Exercises 27 and 28, view vectors in \(\mathbb{R}^{n}\) as \(n \times 1\) matrices. For u and v in \(\mathbb{R}^{n}\), the matrix product \(\mathbf{u}^{T} \mathbf{v}\) is a \(1 \times 1\) matrix, called the scalar product, or inner product, of u and v. It is usually written as a single real number without brackets. The matrix product uvT is an \(n \times n\) matrix, called the outer product of u and v. The products \(\mathbf{u}^{T} \mathbf{v} \text { and } \mathbf{u} \mathbf{v}^{T}\) will appear later in the text.
Let \(\mathbf{u}=\left[\begin{array}{r}-2 \\ 3 \\ -4\end{array}\right]\) and \(\mathbf{v}=\left[\begin{array}{l}a \\ b \\ c\end{array}\right]\)
Compute \(\mathbf{u}^{T} \mathbf{v}, \mathbf{v}^{T} \mathbf{u}, \mathbf{u} \mathbf{v}^{T}\), and \(\mathbf{v u}^{T}\).
Questions & Answers
QUESTION:
In Exercises 27 and 28, view vectors in \(\mathbb{R}^{n}\) as \(n \times 1\) matrices. For u and v in \(\mathbb{R}^{n}\), the matrix product \(\mathbf{u}^{T} \mathbf{v}\) is a \(1 \times 1\) matrix, called the scalar product, or inner product, of u and v. It is usually written as a single real number without brackets. The matrix product uvT is an \(n \times n\) matrix, called the outer product of u and v. The products \(\mathbf{u}^{T} \mathbf{v} \text { and } \mathbf{u} \mathbf{v}^{T}\) will appear later in the text.
Let \(\mathbf{u}=\left[\begin{array}{r}-2 \\ 3 \\ -4\end{array}\right]\) and \(\mathbf{v}=\left[\begin{array}{l}a \\ b \\ c\end{array}\right]\)
Compute \(\mathbf{u}^{T} \mathbf{v}, \mathbf{v}^{T} \mathbf{u}, \mathbf{u} \mathbf{v}^{T}\), and \(\mathbf{v u}^{T}\).
ANSWER:Solution:Step 1:In the given problem we have two vectors in n as n × 1 matrices view. .For u and v in n, the matrix product uTv is a 1 × 1 matrix, called the scalar product, or inner product, of u and v. It is usually written as a single real number without brackets. The matrix product uvT is an n × n matrix, called the outer product of u and v.