Given the transformation $T:\Bbb R^5 \to\Bbb R^2$ where $T(x) = Ax$, how many rows and columns does matrix $A$ have?

Question

Given the transformation $T:\Bbb R^5 \to\Bbb R^2$ where $T(x) = Ax$, how many rows and columns does matrix $A$ have?

Please tell me the procedure to solve this question.

Answer 1

By convention

• $x$ is a $5$-row, $1$-column matrix,
• $A$ is a $2$-row, $5$-column matrix,

allows the multiplication $Ax$ to be defined and

• $Ax$ is a $2$-row, $1$-column matrix.

Parallel to this is the naming, in which rows are said first and columns second. It is said that $x$ is $5\times 1$, $A$ is $2\times 5$, and $Ax$ is a $2\times 1$ matrix.

Answer 2

$T$ maps a $5$-tuple ($R^5$) vector to a $2$-tuple. So $A$ is a $2 \times 5$ matrix. Another way to see this is to remember that the image of $T $ is in $R^2$ and its relationship with columns of $A $ (2-tuples). Similarly kernel of $T $ is a subspace of $R^5$ and the number if columns of $A$ must equal $5$.