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Linear transformations and linear spaces

Represent linear transformations by matrices, find null spaces and ranges, and test subspaces, independence, span and bases.

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  • GCE A-Level H2 Further Mathematics 2027
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Enlarged diagram

A matrix is a compact record of a transformation: its columns say where the basis vectors go, and everything else follows by linearity. This lesson reads a transformation from its matrix, finds what it flattens (its kernel) and what it can reach (its range), connects their dimensions through the rank–nullity relationship, decides whether a set of vectors or polynomials forms a subspace, and composes transformations in the right order.

A matrix records where the basis vectors go

A transformation T:Rⁿ → R^m is linear when T(u + v) = T(u) + T(v) and T(λu) = λ T(u) for all vectors and scalars. Then every vector’s image is fixed by the images of the standard basis vectors, and T(x) = Mx where the columns of the m × n matrix M are T(e₁),…,T(eₙ).

For example, T(x,y) = (x + y, 2y, x-y) maps R² → R³ with T(e₁) = (1,0,1) and T(e₂) = (1,2,-1), so

M = (1, 1; 0, 2; 1, -1), M (3; 1) = (4; 2; 2).

A translation such as T(x) = x + (1,0) is not linear: it sends 0 to (1,0), but every linear map sends 0 to 0.

Doing A first and then B is the single transformation BA: the matrix nearest the vector acts first.

Worked example 1

What a transformation flattens

T:R³ → R² is given by T(x,y,z) = (x + z, y + z). Find its matrix, its null space and its range, and check rank + nullity.

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  1. Matrix and null space

    Method

    M = (1, 0, 1; 0, 1, 1).

    Reason

    The null space is everything sent to 0.

    Working

    Answer

    x + z = 0 and y + z = 0 give (x,y,z) = t(-1,-1,1): a line through the origin, so the nullity is 1.

  2. Range

    Method

    The first two columns (1,0) and (0,1) already span R².

    Reason

    The range is the column space.

    Working

    Answer

    The range is all of R², so the rank is 2, and 2 + 1 = 3 is the dimension of R³.

Linear spaces, subspaces and bases

A linear space (vector space) over R is a set with addition and scalar multiplication satisfying the usual axioms: addition is commutative and associative, there is a zero vector and every vector has a negative, and scalar multiplication distributes over both additions, with 1v = v and (λμ)v = λ(μv). Examples include Rⁿ and the polynomials of degree at most 2.

A non-empty subset of a linear space is a subspace exactly when it contains 0 and is closed under addition and scalar multiplication.

  • Vectors are linearly independent if the only combination giving 0 has all coefficients 0; otherwise one is a combination of the others. (1,2,3), (4,5,6), (7,8,9) are dependent: the middle one is their average.
  • The span of a set is all of its linear combinations.
  • A basis is an independent spanning set; the number of vectors in any basis is the dimension.

Check your understanding 1

Subspace or not?

Decide whether each set is a subspace. For each subspace, give a basis and its dimension.

  1. {(x,y,z):x + y + z = 0} in R³.
  2. {(x,y,z):x + y + z = 1} in R³.
  3. The polynomials p of degree at most 2 with p(1) = 0.
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  1. Yes: it contains 0, and sums and multiples of solutions are solutions. Basis {(1,-1,0),(1,0,-1)}; dimension 2, a plane through the origin.
  2. No: 0 is not in it (0 ≠ 1).
  3. Yes: (p + q)(1) = 0 and (λ p)(1) = 0. Every such p is a(x-1) + b(x²-1), and these two are independent, so the dimension is 2.

Spot the mistake 1

Rotate, then stretch

About 5 min

R = (0, -1; 1, 0) rotates the plane 90° anticlockwise and S = (2, 0; 0, 1) stretches by 2 in the x-direction. A student says “rotate, then stretch” has matrix RS. Explain the error and give the correct matrix.

Write your answer, then compare it with the solution

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  1. Order the product

    Method

    R acts first, so it must be next to the vector: S(Rx) = (SR)x.

    Reason

    Matrix products are generally not commutative.

    Working

    Answer

    SR = (0, -2; 1, 0), RS = (0, -1; 2, 0).

Two orders give different rectangles

The solid rectangle has corners (0, 0), (0, 1), (−2, 1), (−2, 0). The dashed rectangle has corners (0, 0), (0, 2), (−1, 2), (−1, 0). The dotted original is the unit square in the first quadrant.

The solid rectangle has corners (0, 0), (0, 1), (−2, 1), (−2, 0). The dashed rectangle has corners (0, 0), (0, 2), (−1, 2), (−1, 0). The dotted original is the unit square in the first quadrant.The solid rectangle has corners (0, 0), (0, 1), (−2, 1), (−2, 0). The dashed rectangle has corners (0, 0), (0, 2), (−1, 2), (−1, 0). The dotted original is the unit square in the first quadrant.
R rotates 90° anticlockwise; S stretches x by 2. Applying R then S to the unit square gives [−2, 0] × [0, 1]. Applying S then R gives [−1, 0] × [0, 2]. The matrix nearest the vector acts first.
Open full-size graphOpen full-size graph
What to noticeA common coordinate scale preserves the rectangles. The original square has sides of length 1; each transformed area is 2.
Figure data

This table is included in print while it is open.

Figure data: Two orders give different rectangles
Seriesx coordinatey coordinate
R then S00
R then S01
R then S-21
R then S-20
R then S00
S then R00
S then R02
S then R-12
S then R-10
S then R00
Original00
Original10
Original11
Original01
Original00

Common mistakes

  • Writing the images of the basis vectors as the rows of the matrix instead of its columns.
  • Applying a composite SR in the order written; the matrix next to the vector acts first.
  • Claiming a set is a subspace without checking that it contains the zero vector and is closed under addition and scalar multiplication.
  • Giving the dimension of the kernel or range without giving a basis, when the question asks for one.
  • Forgetting that dim(kernel) + dim(range) equals the number of columns.

Before you move on

Check your understanding 2

What does this matrix flatten?

The transformation T:R² → R² has matrix M = (2, 1; 4, 2). Find the dimension of its kernel and of its range, and describe both geometrically.

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M (x; y) = 0 gives 2x + y = 0, so the kernel is the line spanned by (1; -2), of dimension 1. The columns (2; 4) and (1; 2) are parallel, so the range is the line spanned by (1; 2), also of dimension 1. As a check, 1 + 1 = 2, the number of columns. T flattens the whole plane onto the line y = 2x.

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