Linear Algebra I
Skills
Every skill in the course, and how you are doing on each.
Vector spaces, subspaces, sums, and direct sums
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Zero vector
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Scalar multiplication
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Subspace test
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Coordinate axes
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Subspace intersection
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Subspace closure
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Spanning set
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Polynomial space
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Upper triangular
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Nonzero coefficient
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Singleton span
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Existence uniqueness
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Function space
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Containment criterion
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Proper subspace
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Matrix subspaces
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Symmetric matrices
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Sum intersection
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Nested intersections
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Three way direct sum
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Idempotent map
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Idempotent classification
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Complementary projections
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Kernel intersection
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Graph subspace
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Basis extension
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Split surjection
102 more topics with one problem
- Coordinate proof
- Vector addition
- Coordinate calculation
- Zero scalar
- Closure properties
- Coordinate axis
- Zero intersection
- Coordinate equality
- Diagonal subspace
- Homogeneous equation
- Linear constraint
- Solution subspace
- Additive closure
- Union counterexample
- Coefficient construction
- Standard vectors
- Coefficient tracking
- Entrywise closure
- Matrix space
- Line intersection
- Slope subspaces
- Abstract vector space
- Coefficient closure
- Direct sum
- Affine line
- Scalar closure
- Zero vector test
- Evaluation kernel
- Pointwise operations
- Planar directions
- Transversality condition
- Unique coordinates
- Matrix subspace
- Multiplication counterexample
- Trace zero
- Empty family
- Indexed intersection
- Universal quantifier
- Least upper bound
- Greatest lower bound
- Intersection containment
- Finite support
- Sequence space
- Distributivity counterexample
- Strict inclusion
- Zero relation
- Coordinate projection
- Unique splitting
- Coordinate projections
- Diagonal map
- Coordinate planes
- Intersection parameter
- Nonunique decomposition
- Parametric decomposition
- Plane line complement
- Sharp criterion
- Symmetric skew
- Unique decomposition
- Even odd decomposition
- Uniqueness proof
- Idempotent maps
- Zero sum plane
- Direct sum criterion
- Sum map
- Product spaces
- Vertical complement
- Affine parameterization
- Decomposition fiber
- Intersection shift
- Eigenspace splitting
- Linear involution
- Projection formulas
- Distributivity failure
- Lattice inclusion
- Strict counterexample
- Component uniqueness
- Finite support obstruction
- Infinite dimensional
- Standard sequences
- Affine complement
- Two point interpolation
- Contradiction construction
- Subspace union
- Complement existence
- Noncanonical choice
- Idempotent projection
- Range kernel splitting
- Degenerate case
- Line complement classification
- Nested intersection criterion
- Recursive direct sum
- Complement classification
- Graph complements
- Nondistributive diamond
- Subspace lattice
- Commuting involutions
- Eigenspace decomposition
- Hyperplane complements
- Normalized line classification
- Codimension one kernel
- Complement parameterization
- Kernel range decomposition
Linear independence, bases, and dimension
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Linear independence
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Basis definition
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Spanning families
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Kronecker delta
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Linear maps
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Kernel criterion
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Coordinate expansion
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Coordinate map
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Linear equivalence
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Index equivalence
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Basis cardinality
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Finite dimension
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Basis construction
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Dimension formula
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Subspace membership
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Basis completion
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Basis defined map
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Prescribed images
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Basis extraction
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Dimension bound
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Field extension
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Restriction of scalars
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Exchange lemma
83 more topics with one problem
- Indexed families
- Nonzero vectors
- Duplicate vectors
- Injective indexing
- Coordinate uniqueness
- Linear span
- Basis coordinates
- Finsupp coordinates
- Injective reindexing
- Subfamilies
- Pullback independence
- Zero relations
- Preserved independence
- Nonzero vector
- Singleton family
- Scalar multiple
- Span membership
- Finite basis
- Reconstruction formula
- Additivity
- Basis transport
- Mapped basis
- Basis reindexing
- Finrank
- Dimension invariance
- Finite bases
- Basis extensionality
- Basis identification
- Independence and spanning
- Matrix coordinates
- Nonstandard basis
- Change of coordinates
- Shifted basis
- Coordinate system
- Oblique basis
- Three dimensional space
- Basis exchange
- Coordinate criterion
- Sharp necessity
- Basis irredundancy
- Deleted vector span
- Coordinate support
- Selected basis span
- Independent family
- Spanning basis
- Linear extension
- Independent seed
- Kernel span disjointness
- Mapped independence
- Span image
- Basis spanning
- Maximal independence
- Strict extension
- Coordinate kernel
- Deleted basis vector
- Deletion test
- Minimal spanning family
- Basis criterion
- Exact size family
- Adjoined direction
- Dimension increment
- Option indexed family
- Finitely supported sequences
- Infinite dimensional example
- Standard free basis
- Restricted extension
- Spanning superset
- Basis universal property
- Unique linear extension
- Span replacement
- Universal extension
- Maximal independent family
- Field dependent independence
- Dimension theorem
- Infinite bases
- Iterated adjunction
- Strict span chain
- Linear equivalence construction
- Coordinate cocycle
- Operator conjugacy
- Span equality
- Basis classification
- Coordinate transport
Linear maps, kernels, images, and rank-nullity
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Map identities
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Identity map
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Kernel computation
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Range computation
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Linear combinations
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Composition kernel
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Kernel inclusion
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Composition range
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Range inclusion
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Map injectivity
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Map surjectivity
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Map uniqueness
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Left inverse
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Preimage construction
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Right inverse
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Kernel equality
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Range equality
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Finite dimensional
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Rank nullity
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Injective surjective
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Kernel formula
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Subspace preimage
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Matrix product
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Direct sum decomposition
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Hom space cancellation
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Range preservation
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Rank one operator
83 more topics with one problem
- Additive structure
- Kernel foundations
- Linear map zero
- Additive inverses
- Linear map negation
- Linear map subtraction
- Vector differences
- Zero map
- Linear map composition
- Map linearity
- Closure proof
- Kernel subspace
- Coordinate action
- Matrix vector product
- Preimage witnesses
- Range subspace
- Matrix defined map
- Matrix recovery
- Standard coordinates
- Zero propagation
- Preimage witness
- Trivial kernel
- Full range
- Range criterion
- Linear map extensionality
- Map identification
- Pointwise equality
- Sum of maps
- Rank subadditivity
- Sum of linear maps
- Fixed bases
- Matrix representation
- Split monomorphism
- Injective postmap
- Surjective premap
- Coordinate isomorphism
- Inverse linear maps
- Matrix composition
- Nonzero scalar
- Scaled linear map
- Dimension obstruction
- Nontrivial kernel
- Equal dimension
- Change of basis
- Matrix inverses
- Transition matrices
- Range formula
- Composition synthesis
- Kernel range formulas
- Coordinate equivalence
- Matrix reconstruction
- Restricted first isomorphism
- Restricted linear map
- Subspace rank nullity
- Complementary operator
- Kernel range duality
- Square linear system
- Split injection
- Induced linear map
- Kernel range computation
- Restriction corestriction
- Surjective precomposition
- Injective postcomposition
- Kernel preservation
- Dimension count
- Hom space
- Matrix coordinate isomorphism
- Composition extremes
- Rank preservation
- Paired linear map
- Simultaneous measurements
- Unique direct sum
- Operator polynomial
- Block diagonal map
- Product linear map
- Rank nullity equality
- Square zero threshold
- Coupled rank bound
- Cube zero filtration
- Cokernel universal property
- Short exact sequence
- Complex defect
- Exactness criterion
Quotient spaces and universal factorization
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Subspace sum
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Subspace image
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Difference criterion
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Quotient projection
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Representative witness
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Quotient lift
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Representative evaluation
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Nested subspaces
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Commuting diagram
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Nested quotients
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Quotient factorization
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Induced map
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Quotient functor
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First isomorphism theorem
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Quotient equivalence
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Third isomorphism theorem
84 more topics with one problem
- Quotient cosets
- Representative equality
- Quotient membership
- Zero coset
- Canonical map
- Projection surjectivity
- Collapsed subspace
- Coset shift
- Representative independence
- Contrapositive proof
- Coset inequality
- Canonical isomorphism
- Well defined map
- Zero subspace quotient
- Total collapse
- Whole space quotient
- Zero vector space
- Kernel containment
- Quotient extensionality
- Surjective cancellation
- Factor map
- Zero image
- Nonzero scalar cancellation
- Scalar coset
- Zero class
- Universal factorization
- Induced map kernel
- Quotient image
- Induced map range
- Exact kernel
- First isomorphism setup
- Induced injectivity
- Quotient dimension
- Compatibility condition
- Induced evaluation
- Map between quotients
- Composition law
- Quotient map kernel
- Quotient map range
- Image preimage
- Subspace correspondence
- Quotient generation
- Surjectivity criterion
- Factor surjectivity
- Coherence law
- Factor composition
- Quotient chain
- Factor map kernel
- Universal property
- Induced injection
- Quotient range
- Rank identity
- Exact preimage
- Quotient map injectivity
- Compatible map
- Quotient map surjectivity
- Linear isomorphism
- Transported subspace
- Second isomorphism theorem
- Canonical equivalence
- Representative action
- Second isomorphism map
- Quotient of quotient
- Correspondence theorem
- Order isomorphism
- Quotient subspaces
- Composition coherence
- Factor kernel
- Quotient filtration
- Factor map synthesis
- Kernel range invariants
- Quotient factor classification
- Kernel fiber equivalence
- Range comparison
- Quotient by graph
- Intersection kernel
- Quotient crt
- Induced map criteria
- Quotient functoriality
- Meet join preservation
- Quotient lattice correspondence
- Change of complement
- Projection intertwining
- Right inverse classification
Dual spaces, annihilators, and transpose maps
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Functional extensionality
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Dual map
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Kernel annihilator
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Annihilator intersection
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Dual basis
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Functional extension
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Separating functionals
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Double annihilator
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Final a
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Dimension balance
110 more topics with one problem
- Dual map evaluation
- Linear functional
- Map composition
- Dual map identity
- Dual map linearity
- Functional addition
- Zero functional
- Annihilator extremes
- Whole space
- Zero submodule
- Annihilator membership
- Dual submodule
- Pointwise vanishing
- Contravariant composition
- Precomposition
- Dual map kernel
- Kernel characterization
- Range annihilator
- Dual map range
- Annihilator antitone
- Order reversal
- Submodule inclusion
- Annihilator of sum
- Submodule sum
- Arbitrary intersection
- Indexed annihilators
- Submodule supremum
- Functional kernel
- Generated submodule
- Span annihilator
- Annihilator
- Dual restriction
- Restriction kernel
- Basis representation
- Coordinate functional
- Bidual evaluation
- Evaluation linearity
- Scalar action
- Injective dual surjective
- Preimage argument
- Surjective dual injective
- Dual surjectivity iff
- Primal injectivity
- Canonical inclusion
- Evaluation
- Dual composition
- Pointwise evaluation
- Annihilator stability
- Galois closure
- Triple annihilator
- Annihilator coannihilator
- Galois connection
- Order equivalence
- Sum of functionals
- Annihilator injectivity
- Subspace equality
- Annihilator dimension
- Codimension
- Dual range equality
- Functional factorization
- Annihilator equivalence
- Quotient dual
- Quotient pullback
- Separating functional
- Whole space criterion
- Zero annihilator
- Biorthogonality
- Dual basis basis
- Dual spanning
- Functional independence
- Dual map formula
- Idempotent nilpotent
- Faithful dualization
- Witness construction
- Dual retraction
- Section surjection
- Split linear map
- Annihilator range
- Quotient dual dimension
- Quotient dual equivalence
- Dual surjectivity
- Injective composite
- Dual pullback
- Image annihilator
- Injective dimension
- Annihilator grassmann
- Functional hyperplane
- Composition association
- Contravariant functor
- Three map duality
- Dual decomposition
- Operator splitting
- Dual bases
- Dual map coordinates
- Transpose matrix
- Annihilator classification
- Finite codimension
- Bidual reflexivity
- Evaluation naturality
- Duality synthesis
- Restriction quotient
- Exact dual sequence
- Quotient inclusion
- Basis independent rank
- Transpose rank
- Double dual boundary
- Finite subspace
- Finite family annihilators
- Lattice duality
- Dual rank nullity
Alternating forms and determinant structure
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Honors extension
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Final b
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Zero row
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Finite product
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Permutation sign
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Matrix powers
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Determinant multiplicativity
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Row column duality
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Adjugate determinant
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Alternating map
116 more topics with one problem
- Determinant normalization
- Finite index set
- Identity matrix
- Nonempty index set
- Zero matrix
- Diagonal matrix
- Permutation expansion
- Leibniz formula
- Two by two matrix
- Determinant invariance
- Inverse permutation
- Matrix transpose
- Dimension exponent
- Scalar matrix
- Negated matrix
- Parity factor
- Cyclic exchange
- Power addition
- Alternating determinant
- Repeated columns
- Repeated rows
- Transpose duality
- Zero column
- Column additivity
- Multilinearity
- Row additivity
- Column scaling
- Homogeneity
- Row scaling
- Coordinatewise weights
- Simultaneous reindexing
- Transpose invariance
- Column permutation
- Row permutation
- Column operation
- Elementary shear
- Row operation
- Signed products
- Six permutations
- Three by three matrix
- Diagonal product
- Ordered index set
- Upper triangular matrix
- Determinant powers
- Transpose product
- Block triangular matrix
- Determinant product
- Diagonal blocks
- Column independence
- Nonzero determinant
- Row independence
- Determinant vanishing
- Linear dependence
- Double inverse
- Inverse determinant
- Nonsingular inverse
- Outer product
- Rank one matrix
- Transpose zero
- Kernel range
- Matrix linear map
- Unit determinant
- Explicit determinant
- Nonzero minor
- Two by two criterion
- Adjugate matrix
- Cofactor expansion
- Two sided identity
- Dimension minus one
- Product determinant
- Leibniz expansion
- Signed permutation sum
- Transpose expansion
- Determinant exchange
- Scalar associativity
- Three matrix product
- Conjugate powers
- Matrix conjugation
- Transpose conjugation
- Plus minus pair
- Rectangular products
- Sylvester identity
- Block diagonal family
- Blockwise transpose
- Product of determinants
- Determinant polynomial
- Quadratic remainder
- Trace coefficient
- Determinant criterion
- Matrix pairing
- Nondegeneracy
- Dependent family
- Vanishing theorem
- Full permutation action
- High arity alternation
- Zero form
- Alternating bilinear form
- Characteristic two
- Skew symmetry
- Adjugate product
- Row cofactor expansion
- Nonvanishing package
- Invertibility equivalences
- Basis change
- Determinant cocycle
- Cramers rule
- Unique solution
- Alternating form uniqueness
- Normalization
- Singularity detection
- Top form action
- Top form composition
- Translation invariance
- Vandermonde determinant
- Gram determinants
- Independence transfer