Define the (i,j)-minor M_{i,j} of an n×n matrix A.

The (n-1)×(n-1) matrix obtained by deleting row i and column j from A.

Define the (i,j)-cofactor C_{i,j}.

C_{i,j} = (-1)^{i+j} det(M_{i,j}), where M_{i,j} is the (i,j)-minor.

Give the recursive definition of det(A) for an n×n matrix A, expanding along row 1.

det(A) = sum from j=1 to n of a_{1j} C_{1j}, with base case det([a]) = a.

State the cofactor expansion of det(A) along an arbitrary row i.

det(A) = sum from j=1 to n of a_{ij} C_{ij}. Along column j: det(A) = sum from i=1 to n of a_{ij} C_{ij}.

Define the classical adjoint adj(A).

The n×n matrix whose (i,j)-entry is C_{j,i}, i.e. the transpose of the cofactor matrix. Note the index swap.

What does it mean for {v_1,...,v_k} to be linearly independent?

c_1 v_1 + ... + c_k v_k = 0 forces c_1 = ... = c_k = 0. Equivalently, no vector in the set is a linear combination of the others.

What does it mean for a set S to span a vector space V?

Every v in V can be written as a finite linear combination of vectors in S.

Define a basis of a vector space V.

A set that is both linearly independent and spans V.

Define the dimension of a vector space.

The number of vectors in any basis of V. It is well-defined because all bases of V have the same cardinality.

List the three conditions for W contained in V to be a subspace.

(1) 0 is in W; (2) closed under addition: u,v in W implies u+v in W; (3) closed under scalar multiplication: v in W and c a scalar implies cv in W.

Define the kernel of a linear transformation T: V -> W.

ker(T) = {v in V : T(v) = 0}.

Define the null space of a matrix A.

Null(A) = {x : Ax = 0}, the kernel of the map x -> Ax.

Define the column space of an m×n matrix A.

Col(A) = span of the columns of A = {Ax : x in R^n}, a subspace of R^m.

Define the row space of a matrix A.

Row(A) = span of the rows of A = Col(A^T).

Define rank and nullity.

rank(A) = dim(Col(A)) = dim(Row(A)); nullity(A) = dim(Null(A)).

What two properties must T satisfy to be a linear transformation?

T(u+v) = T(u) + T(v) and T(cv) = c T(v) for all u, v and all scalars c. Equivalently T(cu+v) = cT(u) + T(v).

Define an invertible (nonsingular) matrix.

A is invertible if there exists B with AB = BA = I_n. Such a B is unique and written A^{-1}.

Define an invertible linear transformation.

T: V -> W is invertible if there exists T^{-1}: W -> V with T^{-1} composed with T = id_V and T composed with T^{-1} = id_W.

Define one-to-one and onto for T: V -> W.

One-to-one: T(u) = T(v) implies u = v, equivalently ker(T) = {0}. Onto: every w in W equals T(v) for some v in V, equivalently Col(A) = W.

Define an idempotent matrix.

A square matrix A with A^2 = A.

Define the trace of a square matrix.

tr(A) = a_11 + a_22 + ... + a_nn, the sum of the diagonal entries.

Define the coordinate vector [v]_B relative to a basis B = {b_1,...,b_n}.

The unique vector of coefficients (c_1,...,c_n) such that v = c_1 b_1 + ... + c_n b_n.

Define the change-of-basis matrix P_{C<-B}.

The unique matrix satisfying [v]_C = P_{C<-B} [v]_B for all v. Its columns are the B-basis vectors written in C-coordinates.

Define the determinant of an n×n matrix A (non-recursive characterization).

The signed n-dimensional volume of the parallelepiped spanned by the columns of A; equivalently the unique alternating multilinear function of the columns normalized so det(I_n) = 1.

State the Rank-Nullity Theorem.

For A in R^{m×n}: dim(Row(A)) + dim(Null(A)) = n, i.e. rank(A) + nullity(A) = n.

Theorem: dim(Row(A)) versus dim(Col(A)).

They are equal; both equal rank(A).

Theorem: A · adj(A) and adj(A) · A.

A · adj(A) = det(A) I_n and adj(A) · A = det(A) I_n.

Theorem: inverse via the adjugate.

If det(A) is not zero, then A^{-1} = (1/det(A)) adj(A).

Theorem: determinant of a transpose.

det(A^T) = det(A).

Theorem: determinant of a product.

det(AB) = det(A) det(B). Consequently det(A^{-1}) = 1/det(A) and det(kA) = k^n det(A) for n×n A.

Theorem: det(A) = 0 iff the columns of A are linearly dependent.

Columns dependent iff there exists x not zero with Ax = 0 iff Null(A) is not {0} iff A is singular iff det(A) = 0.

Theorem: if ker(T) is not {0} then det(A) = 0. Proof sketch.

Choose v not zero with Av = 0. If A were invertible, v = A^{-1} 0 = 0, a contradiction. So A is not invertible, hence det(A) = 0.

Theorem: if det(A) = 0 then ker(T) is not {0}. Proof sketch.

det(A) = 0 implies A is not invertible, so the reduced row echelon form of A has a free variable, so Ax = 0 has a nontrivial solution, so ker(T) is not {0}.

Theorem: the set of n×n matrices with trace 0 is a subspace. Proof sketch.

(1) tr(0) = 0 so 0 is in S. (2) tr(A+B) = tr(A) + tr(B) = 0 + 0 = 0. (3) tr(cA) = c tr(A) = 0. All three subspace conditions hold.

Theorem: F: P_n(x) -> P_{n+1}(x) given by F(p) = integral from 1 to x of p(t) dt is linear. Proof sketch.

F(p+q) = integral of (p+q) = integral of p + integral of q = F(p) + F(q), and F(cp) = integral of cp = c times integral of p = c F(p). Both linearity conditions hold.

Theorem: adding a multiple of one column to another leaves the determinant unchanged. Proof sketch.

det is multilinear and alternating. Replacing column j by col_j + k col_i splits the determinant into det(..., col_j, ...) + k det(..., col_i, ...). The second determinant has two identical columns, so it vanishes.

Theorem: if A is idempotent then det(A) is 0 or 1. Proof sketch.

A^2 = A implies det(A^2) = det(A), so det(A)^2 = det(A), so d^2 - d = 0, so d(d-1) = 0, giving d = 0 or d = 1.

Theorem: if 8A^{-1} = A^2 for an n×n invertible A, then det(A) = 2^n. Proof sketch.

Multiply by A to get 8I = A^3. Take determinants: 8^n = det(A)^3, so det(A) = 8^{n/3} = 2^n.

Theorem: P_{C<-B} and P_{B<-C} are inverses. Proof sketch.

For every v, [v]_B = P_{B<-C} [v]_C and [v]_C = P_{C<-B} [v]_B. Composing gives [v]_B = P_{B<-C} P_{C<-B} [v]_B for all v, so the product is the identity.

Theorem: det(P_{C<-B}) in terms of the basis matrices.

If B and C are the matrices whose columns are the basis vectors, then P_{C<-B} = C^{-1} B and det(P_{C<-B}) = det(B)/det(C).

Theorem: similar matrices have equal determinants.

If B = P^{-1} A P, then det(B) = det(P^{-1}) det(A) det(P) = det(A).

Theorem: volume scaling under T(x) = Ax on R^3.

For a solid S of volume V, the volume of T(S) is |det(A)| V. In R^n the same holds with n-dimensional volume.

Theorem: volume preservation.

A linear transformation T with standard matrix A preserves volume iff |det(A)| = 1, i.e. det(A) = 1 or det(A) = -1.

Theorem: the rotation matrix has determinant 1.

For A = [[cos t, -sin t], [sin t, cos t]], det(A) = cos^2 t + sin^2 t = 1 for every t. Rotations preserve area and orientation.

Theorem: real eigenvalues of a 2D rotation matrix.

det(A - lambda I) = lambda^2 - 2 cos(t) lambda + 1. The discriminant is -4 sin^2(t), which is at least 0 only when sin(t) = 0, i.e. t = 0 or t = pi, giving lambda = 1 or lambda = -1.

Theorem: eigenvalue criterion.

A real number lambda is an eigenvalue of A iff det(A - lambda I) = 0.

State Cramer's Rule.

If A is n×n and invertible and Ax = b, then x_i = det(A_i)/det(A), where A_i is A with column i replaced by b.

Theorem: if T(e_i) = v_i and S(v_i) = e_i, the standard matrices satisfy B = A^{-1}.

The matrix A of T has columns v_1, v_2, v_3; S reverses the mapping, so its standard matrix is the inverse of A.

Theorem: the columns of an invertible n×n matrix form a basis of R^n. Proof sketch.

A invertible implies det(A) is not zero, so the columns are linearly independent; there are n of them in R^n, so they also span R^n. Independent and spanning means basis.

Theorem: dim(Null(A)) - dim(Null(A^T)).

For A in R^{m×n}, dim(Null(A)) - dim(Null(A^T)) = n - m.