Linear Transformations

A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.

The standard form of a linear function is given by $$y = mx + b$$ , where $$m$$ is the slope and $$b$$ is the y-intercept.

The slope of a linear function represents the rate of change of the function; it indicates how much the output value changes for a unit change in the input value.

The y-intercept is the point where the graph of the function crosses the y-axis, represented by the value of $$b$$ in the equation $$y = mx + b$$ .

A vertical stretch multiplies the output of the function by a factor greater than 1, making the graph steeper.

A vertical compression multiplies the output of the function by a factor between 0 and 1, making the graph less steep.

A reflection is a transformation that flips a graph over a specified line, such as the x-axis or y-axis.

A horizontal shift moves the graph of the function left or right without changing its shape.

A vertical shift moves the graph of the function up or down without changing its slope.

The parent function of linear functions is $$y = x$$ , which is a straight line with a slope of 1 and a y-intercept of 0.