Common Core · High school
HSN-VM.C.11
High school — Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector; work with matrices as transformations of vectors.
A note on this mapping: HSN-VM.C.11 is a (+) standard — beyond the college- and career-ready threshold, i.e. precalculus rather than Algebra II. Reading Ax as a combination of the columns of A is this same product, seen from the other side.
The 5 skills that cover it
- Linear CombinationsBuild one vector out of others: evaluate a combination such as 3u - 2v, work backwards to find the weights that reach a target, decide whether a target can be reached at all, and see that a matrix times a vector is nothing more than a combination of the columns.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 →
- Linear TransformationsA function that moves every vector at once, and the matrix that describes it: test the two rules a linear map obeys, build the standard matrix from the images of the basis vectors, recognise rotations, reflections, scalings and shears, and read a composition as a matrix product.Read the lesson →
- Kernel, Range & Rank–NullityWhat a transformation loses and what it can reach: decide whether a vector is sent to zero, find the dimension of the kernel and a vector that spans it, decide whether a target is in the range, and use rank plus nullity to account for every dimension of the input.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 →
What a student should be able to do
- Evaluate a linear combination of two or three vectors.
- Find the weights that write one vector as a combination of others, by solving the system they give.
- Decide whether a vector can be written as a combination of a given set, and say why not when it cannot.
- Read a matrix-vector product as a linear combination of the columns of the matrix.
- 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.
- Decide whether a map is linear, and name the rule that fails when it is not.
- Find the image of a vector under a transformation given by a matrix.
- Build the standard matrix of a transformation from the images of the basis vectors.
- Recognise the matrix of a rotation, reflection, scaling or shear, and compose two transformations by multiplying their matrices.
- Decide whether a vector lies in the kernel of a transformation, and find a vector that spans the kernel.
- Find the dimension of the kernel and of the column space of a matrix.
- Decide whether a vector lies in the range of a transformation.
- Apply the rank-nullity relation, and connect a kernel of only the zero vector to a one-to-one map.
- 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.