Linear Transformations, Rank and Nullity: every key term you need (+ practice quiz)
25 flashcard terms for Linear Algebra Topic 5, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
A map between vector spaces that preserves addition and scalar multiplication, so the image of a linear combination is the same combination of the images.
Domain and codomain
The space a transformation accepts inputs from and the space its outputs are declared to lie in; the codomain need not coincide with the set of values actually attained.
Image of a transformation
The set of vectors actually produced as outputs. It is a subspace of the codomain and coincides with the column space when the map is given by a matrix.
Kernel of a transformation
The set of inputs sent to the zero vector. It is a subspace of the domain and equals the null space of any matrix representing the map.
Standard matrix of a transformation
For a map on coordinate space, the matrix whose columns are the images of the standard basis vectors; it reproduces the transformation by multiplication.
Matrix representation in general bases
Given ordered bases of the domain and codomain, the matrix whose columns hold the coordinates of the images of the domain basis vectors relative to the codomain basis.
Injective transformation
A transformation sending distinct inputs to distinct outputs. For a linear map this happens exactly when the kernel contains only the zero vector.
Surjective transformation
A transformation whose image is all of the codomain. For a matrix map this holds exactly when the matrix has a pivot in every row.
Isomorphism
A linear transformation that is both injective and surjective. Spaces linked by one are structurally identical and necessarily have the same dimension.
Rank
The dimension of the image of a transformation, equivalently the number of pivot columns of a representing matrix and the dimension of its column space.
Nullity
The dimension of the kernel, equal to the number of free variables in the associated homogeneous system and to the number of nonpivot columns.
Rank nullity theorem
For a linear map from a space of dimension n, the rank plus the nullity equals n. Every input direction is either collapsed or contributes to the image.
Row rank equals column rank
The theorem that the dimension of the row space equals the dimension of the column space for any matrix, so rank may be computed from either point of view.
Full rank matrix
A matrix whose rank equals the smaller of its two dimensions. For a square matrix this is one more restatement of invertibility.
Rank of a product
The rank of a product is at most the rank of either factor, because the image of the product sits inside the image of the left factor.
Composition of transformations
Applying one linear map after another yields a linear map whose matrix is the product of the individual matrices, written in the order the maps are applied from the right.
Projection map
A linear transformation equal to its own square, so applying it twice changes nothing further; it splits the space into its image and its kernel.
Rotation transformation
A map of the plane turning every vector through a fixed angle about the origin. It preserves lengths and angles and its matrix has orthonormal columns.
Shear transformation
A map that slides points parallel to a fixed line by an amount proportional to their distance from that line, leaving areas and one direction unchanged.
Dilation and contraction
Maps scaling every vector by a fixed positive factor, expanding when the factor exceeds one and shrinking when it lies between zero and one.
Similar matrices
Two square matrices related by conjugation with an invertible matrix. They represent one transformation in different bases and share rank, trace and determinant.
Linearity conditions
The two requirements defining a linear map: additivity on sums and homogeneity on scalar multiples. Together they force the zero vector to map to the zero vector.
Left inverse of a transformation
A map undoing a transformation on its domain. It exists exactly when the transformation is injective, which for matrices means independent columns.
Right inverse of a transformation
A map that the transformation undoes on the codomain. It exists exactly when the transformation is surjective, that is when the columns span the codomain.
Space of linear maps
The collection of all linear transformations between two fixed spaces, itself a vector space under pointwise operations, with dimension equal to the product of the two dimensions.