Common Core · High school
HSN-VM.C.10
High school — Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers; the determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
A note on this mapping: HSN-VM.C.10 is a (+) standard — beyond the college- and career-ready threshold, i.e. precalculus rather than Algebra II. HSN-VM.C.10 names the determinant and the existence of an inverse, which is what independence looks like for n vectors in n dimensions. Independence as a property of any set of vectors, and the dependence relation that comes with it, is first-year university content; the Common Core stops before it.
The 7 skills that cover it
- Linear IndependenceWhether a set carries any repetition: test two or three vectors for independence, find the dependence relation when there is one, spot the value that makes a set dependent, read independence off a pivot count, and know why more than n vectors in n dimensions can never be independent.Read the lesson →
- Basis, Dimension & CoordinatesThe smallest set that still reaches everywhere: decide whether a set is a basis, count the dimension of a span or a subspace, write a vector in coordinates relative to a chosen basis, and turn those coordinates back into the ordinary vector.Read the lesson →
- DeterminantsOne number that says whether a matrix flattens space: compute a 2 by 2 and a 3 by 3 determinant, use row operations and a triangular form to make it quick, apply the rules for a product, a scalar multiple and a transpose, read the determinant as an area, and solve a small system with it.Read the lesson →
- Inverse MatricesThe matrix that undoes another: invert a 2 by 2 with the formula, decide from a determinant whether an inverse exists at all, find one by reducing A beside the identity, solve a system with it, and collect the many equivalent ways of saying a matrix is invertible.Read the lesson →
- Rank & the Shape of SolutionsCounting what a matrix really pins down: read the rank off a reduced form, work out the nullity from it, count free variables, compare the rank of a matrix with the rank of its augmented form to decide consistency, and say the largest rank a given shape allows.Read the lesson →
- Eigenvalues & EigenvectorsThe directions a matrix leaves pointing the same way: check whether a vector is an eigenvector and read its eigenvalue, form the characteristic equation of a 2 by 2, find both eigenvalues and an eigenvector for each, and use the trace and determinant as a shortcut.Read the lesson →
- Diagonalization & Its UsesChoosing the basis that makes a matrix simple: decide whether a matrix can be diagonalised, write down the diagonal matrix of eigenvalues, raise a matrix to a high power through its eigenbasis, and read the long run of a two-state chain or a population model off the eigenvalues.Read the lesson →
What a student should be able to do
- Decide whether a set of vectors is linearly independent, and justify the answer.
- Find the weights of a dependence relation when a set is dependent.
- Find the value of a parameter that makes a set dependent.
- Read independence off the pivot positions, and give the largest independent set n dimensions allow.
- Decide whether a set of vectors is a basis for a space, and say which of the two conditions fails.
- Find the dimension of a span or a subspace, and how many vectors a basis for it needs.
- Find the coordinates of a vector relative to a given basis.
- Convert coordinates back to the standard vector, and read an entry of a change-of-coordinates matrix.
- Compute the determinant of a 2 by 2 matrix and of a 3 by 3 matrix by cofactor expansion.
- Use row operations or a triangular form to compute a determinant, and apply the rules for det(AB), det(kA) and det of a transpose.
- Find the value of a parameter that makes a determinant zero.
- Use a determinant as the area of a parallelogram or triangle, and solve a 2 by 2 system by Cramer’s rule.
- Find the inverse of a 2 by 2 matrix, and decide from the determinant whether a matrix has one.
- Find an entry of an inverse by reducing the matrix beside the identity.
- Solve a matrix equation using an inverse.
- Apply the rule for the inverse of a product, and recognise equivalent statements of invertibility.
- Find the rank of a matrix from its reduced row echelon form, and the number of free variables.
- Use the rule nullity = n - rank to find one of the three from the other two.
- Compare the rank of a coefficient matrix with the rank of the augmented matrix to decide consistency.
- Give the largest possible rank of a matrix of a stated shape, and what row operations do to rank.
- Decide whether a vector is an eigenvector of a matrix, and give the eigenvalue that goes with it.
- Form the characteristic equation of a 2 by 2 matrix and solve it for both eigenvalues.
- Find an eigenvector for a given eigenvalue.
- Use the trace and determinant as the sum and product of the eigenvalues, and give the eigenvalues of a triangular matrix.
- Decide whether a matrix is diagonalisable, and say why when it is not.
- Give the diagonal matrix D in A = PDP⁻¹, and an entry of a matrix power computed through it.
- Find the steady-state vector of a two-state chain.
- Say which eigenvalue governs the long-run behaviour of a model, and what the eigenvalues mean in it.