Linear transformations · Definition

Linear transformation

linear algebralinear transformations

Before you start you need to know

Linear transformations are the functions that respect the structure of a vector space; they describe rotations, reflections, projections and changes of scale.

Definition

A linear transformation \(T\) is a function between vector spaces that preserves addition and scalar multiplication.

\[T(u+v)=T(u)+T(v),\quad T(cu)=cT(u)\]

The two conditions can be merged into one: \(T(au+bv)=aT(u)+bT(v)\). In words, transforming a linear combination is the same as combining the transforms. An immediate consequence is \(T(0)=0\), so a function that moves the origin, such as a translation, is never linear.

Linearity means \(T\) is determined by what it does on a basis. If \(v=a_1v_1+\cdots+a_nv_n\), then \(T(v)=a_1T(v_1)+\cdots+a_nT(v_n)\). For instance, if \(T(1,0)=(2,1)\) and \(T(0,1)=(-1,3)\), then \(T(x,y)=x(2,1)+y(-1,3)\) for every vector.

Examples: \(T(x,y)=(x+y,x-y)\) is linear; differentiation, \(p\mapsto p'\), is linear on the space of polynomials; multiplication by a matrix, \(x\mapsto Ax\), is linear on \(\mathbb{R}^n\). The maps \(x\mapsto x^2\), \(x\mapsto x+1\) and the absolute value are not. Later we will see that every linear transformation between finite-dimensional spaces can be written as a matrix.

Typical mistake: calling a function «linear» because its graph is a line, like \(f(x)=2x+1\). In linear algebra it must satisfy \(f(0)=0\), and this one does not. To disprove linearity a single counterexample suffices: check \(T(0)=0\) or test additivity on two concrete vectors.

Think about it

Is \(f(x)=2x+1\) linear as a function from \(\mathbb{R}\) to \(\mathbb{R}\)?

No: \(f(0)=1\ne0\). It is an affine function, not linear in the sense of linear algebra.

If you know \(T(v_1)\) and \(T(v_2)\) for a basis \(\{v_1,v_2\}\) of \(\mathbb{R}^2\), can you compute \(T\) of any vector?

Yes: every vector is \(a_1v_1+a_2v_2\), and by linearity \(T(v)=a_1T(v_1)+a_2T(v_2)\).

Exercises 10

  1. Let \(T(x,y)=(2x,3y)\). What is \(T(1,1)\)?
  2. Which of these functions from \(\mathbb{R}^2\) to \(\mathbb{R}^2\) is linear?
  3. If \(T\) is a linear transformation, what is \(T(0)\)?
  4. Let \(T(x,y)=(x+y,\,x-y)\). Compute \(T(2,3)\).
  5. A linear transformation \(T\) satisfies \(T(1,0)=(2,1)\) and \(T(0,1)=(-1,3)\). Compute \(T(3,2)\).
  6. A linear transformation satisfies \(T(1,1)=(3,1)\) and \(T(1,-1)=(1,5)\). What is \(T(1,0)\)?
  7. A linear transformation satisfies \(T(u)=(1,2)\) and \(T(v)=(3,-1)\). Compute \(T(2u-v)\).
  8. Which of these geometric operations on the plane is NOT a linear transformation?
  9. Let \(P\) be the space of polynomials. Which of these functions from \(P\) to \(P\) or to \(\mathbb{R}\) is linear?
  10. A linear transformation \(T\) satisfies \(T(1,2)=(0,0)\) and \(T(1,0)=(1,1)\). Compute \(T(0,1)\).

References

  • Gilbert Strang, Introduction to Linear Algebra, 6th edition, Wellesley-Cambridge Press, 2023. §8.1
  • Sheldon Axler, Linear Algebra Done Right, 4th edition, Springer, 2024. §3.A
  • David C. Lay, Steven R. Lay, Judi J. McDonald, Linear Algebra and Its Applications, 6th edition, Pearson, 2021. §1.8 y §1.9