Common Core · High school
HSN-VM.C.12
High school — Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
A note on this mapping: HSN-VM.C.12 is a (+) standard — beyond the college- and career-ready threshold, i.e. precalculus rather than Algebra II.
The 3 skills that cover it
- 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 →
- 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 →
What a student should be able to do
- 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.
- 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.