Vector spaces and subspaces

50 problems · Subspaces, linear independence, span, bases and dimension.

Search problems Level

Ex.285Checking the vector space axioms

#vector space axioms #examples #counterexamples

High school

Ex.319A plane as a subspace

#subspaces #basis

Vector spaces and subspaces

Ex.590Recognizing bases

#basis #dimension #canonical basis

Vector spaces and subspaces

High school

Ex.594Linear independence

#linear independence #definition #examples

Vector spaces and subspaces

High school

Ex.595Span of a set of vectors

#span #linear combinations #generating set

Vector spaces and subspaces

High school

Ex.596The subspace criterion

#subspace criterion #examples

Vector spaces and subspaces

High school

Ex.597Which subsets are subspaces?

#subspace criterion #counterexamples

Vector spaces and subspaces

High school

Ex.1194A basis of a hyperplane in $\mathbb{R}^4$

#basis #hyperplane #dimension

Vector spaces and subspaces

Ex.1195Image of a basis under a linear map

#basis #image #linear map

Vector spaces and subspaces

Ex.1196A basis of a polynomial space

#polynomial spaces #basis #dimension

Vector spaces and subspaces

Ex.1197The canonical basis and matrix representation

#canonical basis #matrix representation #linear map

Vector spaces and subspaces

Ex.1199Eigenspaces as subspaces

#eigenspaces #subspaces #dimension

Vector spaces and subspaces

Ex.1200Dimension of a subspace

#dimension #subspace #equations

Vector spaces and subspaces

Ex.1205Independence depending on a parameter

#linear independence #parameter #determinant

Vector spaces and subspaces

Ex.1206Isomorphism criterion

#isomorphism #dimension #injective linear map

Vector spaces and subspaces

Ex.1207Basis of a kernel

#kernel #basis #linear map

Vector spaces and subspaces

Ex.1208Kernel and image of a linear map

#kernel #image #linear map #rank-nullity

Vector spaces and subspaces

Ex.1209Kernel of a differentiation-like map

#kernel #differentiation #polynomials

Vector spaces and subspaces

Ex.1216The space of linear maps

#linear maps #vector space structure #dimension

Vector spaces and subspaces

Ex.1217Injective linear map and kernel

#injectivity #kernel #linear map

Vector spaces and subspaces

Ex.1218Column space and row space

#column space #row space #rank

Vector spaces and subspaces

Ex.1219Kernel of a 3×3 matrix and basis

#kernel #basis #matrix #rank-nullity

Vector spaces and subspaces

Ex.1220Subspace generated by a family

#span #generated subspace #dimension

Vector spaces and subspaces

Ex.1221Subspace generated by columns of a matrix

#column space #generators #rank

Vector spaces and subspaces

Ex.1222Subspace inclusion and dimension

#inclusion #dimension #equality

Vector spaces and subspaces

Ex.1223The direct sum criterion

#direct sum #criterion #proof

Vector spaces and subspaces

Ex.1224Surjective linear map and image

#surjectivity #image #linear map

Vector spaces and subspaces

Ex.1896Completing a family into a basis

#basis #completion #incomplete basis theorem

Vector spaces and subspaces

Ex.1897Change of basis in a polynomial space

#change of basis #transition matrix #polynomials

Vector spaces and subspaces

Ex.1898Existence of a supplementary subspace

#supplementary subspace #basis extension #existence

Vector spaces and subspaces

Ex.1899Hyperplanes and linear forms

#hyperplane #linear form #kernel #dimension

Vector spaces and subspaces

Ex.1900Dimension of a quotient space

#quotient space #dimension #proof

Vector spaces and subspaces

Ex.1901Direct sum of subspaces

#direct sum #supplementary subspaces #dimension

Vector spaces and subspaces

Ex.1902Decomposing a vector in a direct sum

#direct sum #decomposition #projection

Vector spaces and subspaces

Ex.1906The dual basis

#dual space #dual basis #linear forms

Vector spaces and subspaces

Ex.1907Eigenspaces sum to the whole space

#eigenspaces #direct sum #diagonalization

Vector spaces and subspaces

Ex.1909Orthonormal basis of a subspace

#Gram-Schmidt #orthonormal basis #subspace

Vector spaces and subspaces

Ex.1915Intersection of two kernels

#intersection #kernel #dimension

Vector spaces and subspaces

Ex.1916Intersection and sum of subspaces

#intersection #sum #dimension formula

Vector spaces and subspaces

Ex.1920Matrix of a linear map in a given basis

#matrix representation #basis #linear map

Vector spaces and subspaces

Ex.1921Rank-nullity on a 3×4 matrix

#rank-nullity #rank #matrix

Vector spaces and subspaces

Ex.1922Orthogonal complement

#orthogonal complement #dimension #inner product

Vector spaces and subspaces

Ex.1923A subspace of polynomials with a root condition

#polynomial subspace #evaluation #dimension

Vector spaces and subspaces

Ex.1924Quotient vector space

#quotient space #cosets #dimension

Vector spaces and subspaces

Ex.1927A subspace depending on a parameter

#parameter #subspace #dimension

Vector spaces and subspaces

Ex.1928Sum of two planes in $\mathbb{R}^3$

#sum #intersection #planes

Vector spaces and subspaces

Ex.1929Symmetric and skew-symmetric matrices

#symmetric #skew-symmetric #direct sum

Vector spaces and subspaces

Ex.1930The subspace of traceless matrices

#trace #subspace #dimension #matrices

Vector spaces and subspaces

Ex.1931Kernel of the transpose

#transpose #kernel #orthogonal complement #rank

Vector spaces and subspaces

Ex.2222Proving the dimension formula

#dimension formula #proof #intersection and sum

Vector spaces and subspaces