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John says that the equation y = 5 is a constant function which is also a linear function. What is your opinion on this statement? Explain.
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It depends on what you mean by “linear.”
- In elementary/coordinate-geometry language (functions of the form y = mx + b), y = 5 is linear: it has slope m = 0 and y-intercept b = 5. Its graph is a straight (horizontal) line.
- In linear algebra the term “linear” means a linear map (homogeneous linear function) satisfying f(x+y)=f(x)+f(y) and f(cx)=c f(x). Under that definition y = 5 is not linear, because it does not send 0 to 0 (f(0)=5 ≠ 0) and it fails additivity (f(1+1)=5 but f(1)+f(1)=10). Functions of the form y = mx + b with b ≠ 0 are called affine functions.
So: y = 5 is a constant (and affine) function; it is called “linear” in basic algebra/graphing, but not a linear transformation in linear algebra.
- In elementary/coordinate-geometry language (functions of the form y = mx + b), y = 5 is linear: it has slope m = 0 and y-intercept b = 5. Its graph is a straight (horizontal) line.
- In linear algebra the term “linear” means a linear map (homogeneous linear function) satisfying f(x+y)=f(x)+f(y) and f(cx)=c f(x). Under that definition y = 5 is not linear, because it does not send 0 to 0 (f(0)=5 ≠ 0) and it fails additivity (f(1+1)=5 but f(1)+f(1)=10). Functions of the form y = mx + b with b ≠ 0 are called affine functions.
So: y = 5 is a constant (and affine) function; it is called “linear” in basic algebra/graphing, but not a linear transformation in linear algebra.
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