A linear map from R^7 to R^5 has rank 4. Find its nullity.
Answer and explanation
3
Rank-nullity uses the 7-dimensional domain: 7−4=3.
Subjects
Linear algebra
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Sample exercises
Show that vectors (1,0) and (1,1) form a basis of R².
Answer and explanation
A linear combination a(1,0)+b(1,1)=(a+b,b) can represent every (x,y) using b=y,a=x−y. The zero combination forces a=b=0.
Credit spanning and independence, or a nonzero determinant argument.
Which set is a linear subspace of R²?
- The unit circle
- All (x,y) with x+y=0
- All points with x>0
- All (x,y) with x+y=1
Answer and explanation
All (x,y) with x+y=0
The homogeneous solution set contains zero and is closed under linear combinations.
What is the rank of [[1,2],[2,4]]?
Answer and explanation
1
The second row is twice the first and the matrix is not zero.
Can three vectors be linearly independent in R²?
- Yes, if all entries are nonzero
- Yes, if all lengths differ
- Only when perpendicular in pairs
- No
Answer and explanation
No
No independent set has more vectors than the dimension.
Explain why the pivot columns of a row-reduced matrix identify basis columns in the original matrix rather than replacing them.
Answer and explanation
Row operations preserve linear relations among columns, so pivot positions select independent original columns spanning the original column space. The reduced columns may live in a different column space.
Credit the distinction between selecting original columns and using transformed columns.
Study notes
Bases, rank and linear maps
A basis both spans a vector space and is linearly independent.
UniversityBases, rank and linear maps · revision notes
A basis both spans a vector space and is linearly independent.
UniversityEigenvectors and diagonalization
Av=λv requires a nonzero vector v.
UniversityEigenvectors and diagonalization · revision notes
Av=λv requires a nonzero vector v.
UniversityOrthogonality and least-squares fitting
The closest point in a subspace is found by an orthogonal residual.
UniversityOrthogonality and least-squares fitting · revision notes
The closest point in a subspace is found by an orthogonal residual.
UniversityDeterminant of a 3 × 3 triangular matrix
Off-diagonal entries do not enter this special-case determinant; this is not true for every matrix.
UniversityDeterminant of a 3 × 3 triangular matrix · formula reference
Off-diagonal entries do not enter this special-case determinant; this is not true for every matrix.
UniversityEigenvalue from a known eigenvector
Verify the vector is nonzero. Distinct eigenvectors can have distinct eigenvalues.
UniversityEigenvalue from a known eigenvector · formula reference
Verify the vector is nonzero. Distinct eigenvectors can have distinct eigenvalues.
UniversityScalar coefficient in an orthogonal projection
The coefficient multiplying u differs from the scalar projection onto the unit vector u/||u||.
UniversityScalar coefficient in an orthogonal projection · formula reference
The coefficient multiplying u differs from the scalar projection onto the unit vector u/||u||.
UniversityRank-nullity for a finite-dimensional map
The dimension is that of the domain, not the codomain. Rank cannot exceed the domain dimension.
UniversityRank-nullity for a finite-dimensional map · formula reference
The dimension is that of the domain, not the codomain. Rank cannot exceed the domain dimension.
UniversitySolve a nonsingular 2 × 2 system
The coefficient determinant is nonzero by construction.
UniversitySolve a nonsingular 2 × 2 system · formula reference
The coefficient determinant is nonzero by construction.
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