Question

Row and column vectors \(\vec{u}\) and \(\vec{v}\) are defined. Find the product \(\vec{u} \vec{v},\) where possible. $$ \vec{u}=\left[\begin{array}{lll} 1 & 2 & 3 \end{array}\right] \quad \vec{v}=\left[\begin{array}{l} 3 \\ 2 \end{array}\right] $$

Short answer

The product \(\vec{u} \vec{v}\) is not possible due to incompatible dimensions.

Step-by-step solution

Identify the Dimensions

To find whether the product is possible, we need to identify the dimensions of the vectors. The vector \(\vec{u}\) is a row vector with dimensions \((1 \times 3)\). The vector \(\vec{v}\) is a column vector with dimensions \((2 \times 1)\). We observe that the inner dimensions do not match, so the multiplication \(\vec{u} \vec{v}\) is not possible.

Conclusion on Multiplicability

Since the inner dimensions of the vectors \(\vec{u}\) and \(\vec{v}\) do not match (3 in \(\vec{u}\) and 2 in \(\vec{v}\)), we conclude that their product cannot be computed using standard matrix multiplication rules.

Key concepts

Row Vector

A row vector is a type of matrix with only one row and multiple columns. This means it is written horizontally. In this case, \[\vec{u} = \left[ \begin{array}{ccc} 1 & 2 & 3 \end{array} \right]\]is an example of a row vector with three elements, which can also be thought of as a 1 x 3 matrix. This format is common for representing a sequence of data points or a single row of values from a larger matrix. It is key in linear algebra operations like matrix multiplication, because its orientation - as rows - affects how we perform operations with other matrices or vectors.

Column Vector

In contrast to a row vector, a column vector is arranged vertically. It has a single column and multiple rows, resembling a column of numbers. An example of this is \[\vec{v} = \left[ \begin{array}{c} 3 \ 2 \end{array} \right]\]which is a 2 x 1 matrix. Column vectors frequently signify dimensional components, like direction or magnitude in physics. Recognizing the orientation and dimensions of column vectors is crucial in operations like matrix multiplication, ensuring each mathematical process respects the vector's layout.

Vector Dimensions

The dimensions of a vector give valuable information about its shape and size. Specifically, they indicate the number of rows and columns. For instance, a row vector with three elements \(\vec{u} = \left[ \begin{array}{ccc} 1 & 2 & 3 \end{array} \right]\) has dimensions 1 x 3, while the column vector \(\vec{v} = \left[ \begin{array}{c} 3 \ 2 \end{array} \right]\) is 2 x 1. Understanding these dimensions ensures proper handling of matrix operations. The dimensions help determine if matrices or vectors can be multiplied by indicating if their inner dimensions are compatible.

Multiplicability of Matrices

Matrix multiplication requires strict adherence to certain rules concerning their dimensions. For two matrices, or vectors, to be multiplicable, the number of columns in the first matrix must equal the number of rows in the second matrix. For example, if we have a row vector \(\vec{u}\) with dimensions 1 x 3, and a column vector \(\vec{v}\) with dimensions 2 x 1, they cannot be multiplied as these inner dimensions (3 and 2) do not match.

• The product is only possible if the first matrix's columns match the second's rows.
• This product will result in a new matrix having dimensions defined by the outer dimensions: 1 x 1 in a compatible scenario, indicating a single numerical result.

Recognizing these rules is essential for determining the feasibility of multiplication, setting the stage for correct matrix algebra.