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How many rows and columns must a matrix $A$ have in order to define a mapping from $\mathbb{R}^{4}$ into $\mathbb{R}^{5}$ by the rule $T(\mathbf{x})=A \mathbf{x} ?$

5 Rows and 4 Columns.

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 8

Introduction to Linear Transformations

Introduction to Matrices

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So we're dealing with a function defined by the cruelty of X equals a X, and we want this function to be a mapping. Ah, from our four and two are five. So we want to start off with the metric, um, a vector of length for and end up with the vector of length five. So we needed to determine the size of a So say, this is our matrix A. We need to determine how many rows and how many columns we need. So the number of rose is determined by the CO domain, which is the range. And so the order of the range has five. Because this is our range and the number of columns is determined by the order of the domain, which is we're starting off with, Ah, a domain of order for So that tells me that we need to have five rows and four columns. So a is going to be a, uh, five by four matrix

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