Section 2.8 Unit 2 highlights
Big question. Given a linear map, what does it produce, what does it forget, and how do we solve equations involving it?
Notation. \(A\mathbf{x}=\mathbf{b}\text{,}\) \([A\mid \mathbf{b}]\text{,}\) \([A\mid \mathbf{0}]\text{,}\) \(\operatorname{rref}([A\mid \mathbf{b}])\) for solving systems, \(\operatorname{rref}(A)\) for studying the matrix, \(\operatorname{col}(A)\text{,}\) \(\operatorname{im}(A)\text{,}\) \(\operatorname{null}(A)\text{,}\) \(\operatorname{rank}(A)\text{,}\) \(\operatorname{nullity}(A)\text{,}\) \(A^{-1}\text{,}\) \(\det(A)\text{.}\)
Concepts. System of linear equations, row reduction, reachable output, null space, image space, column space, span, linear independence, basis, coordinates, dimension, rank, nullity, inverse, determinant, redundant feature.
Results. The equation \(A\mathbf{x}=\mathbf{b}\) is consistent exactly when \(\mathbf{b}\) is in the column space. Nonzero null-space directions create different inputs with the same output. Rank counts independent output directions. Nullity counts independent forgotten input directions. The rank-nullity theorem says
\begin{equation*}
\operatorname{rank}(A)+\operatorname{nullity}(A)=n
\end{equation*}
for an \(m\times n\) matrix. A square matrix is invertible precisely when no nonzero input direction is forgotten; equivalently, it has full rank and nonzero determinant.
Toolbox skills. U2-LO1 through U2-LO7 use augmented matrices, elementary row operations, REF and RREF, homogeneous and nonhomogeneous systems, null-space and column-space bases, rank, nullity, independence tests, inverse computation, cofactor expansion, determinant properties, and determinant tests.
Main applications. Systems from geometry, lines and planes as solution sets, small parameter-fitting systems, intersections of planes, redundant data features, difference matrices, resource constraints, and output layers that cannot detect null-space directions.
Connections. Unit 1 supplies matrices, matrix-vector products, geometric matrix actions, and linear maps. Unit 3 uses systems and normal vectors again for tangent planes. Unit 4 replaces unreachable targets by least-squares approximations and proves the \(A^TA\) invertibility fact used in normal equations. Unit 6 uses rank and null-space ideas when SVD separates dominant and forgotten directions.
Study anchors. U2-LO1 and U2-LO2: Linear systems. U2-LO3 and U2-LO6: Understanding a linear map: a preview; Rank of a matrix; Applications and computation recap; linked notebook. U2-LO4: Subspaces; Independence, bases, and rank. U2-LO5 and U2-LO7: Inverses and determinants.
Applications and computations readiness checklist. Given a short Unit 2 calculation, code snippet, table, or diagram, I can decide whether a target is reachable; read an rref output of an augmented matrix and separate variable columns from the right-hand side; identify pivot and free variables; interpret \(A\mathbf{z}=\mathbf{0}\) as a forgotten direction; explain why \(A(\mathbf{x}+\mathbf{z})=A\mathbf{x}\text{;}\) read
matrix_rank(A) as transmitted dimension; use determinant and rank to test invertibility; and explain when different parameter vectors give the same output.
Common mistakes. Treating row reduction as the goal rather than a tool; confusing row space and column space; using columns of rref instead of pivot columns of the original matrix for \(\operatorname{col}(A)\text{;}\) assuming every spanning list is a basis; forgetting that a nonzero null vector means non-unique inputs; calling a singular square matrix reversible.
