linear algebra test 2

Determinant

A scalar value assigned to a square matrix that indicates whether it is invertible and characterizes geometric scaling.

Determinant of a $2 \times 2$ matrix with row 1 $(a, b)$ and row 2 $(c, d)$

$$ad - bc$$

Methods to compute a $3 \times 3$ determinant

Cofactor expansion along any row or column, or row reduction to triangular form while tracking determinant operations.

Minor $M_{ij}$

The determinant of the submatrix obtained by deleting row $i$ and column $j$ from a square matrix.

Formula relating cofactor $C_{ij}$ to minor $M_{ij}$

$$C_{ij} = (-1)^{i+j} M_{ij}$$

Cofactor expansion

A method to calculate a determinant by taking the sum of the entries of any single row or column multiplied by their corresponding cofactors.

Checkerboard sign pattern of cofactors for a $3 \times 3$ matrix

Row 1: $(+, -, +)$; Row 2: $(-, +, -)$; Row 3: $(+, -, +)$.

Optimal row or column choice for cofactor expansion

The row or column containing the greatest number of zero entries.

Upper triangular matrix

A square matrix where all entries strictly below the main diagonal are equal to zero.

Determinant of a triangular matrix

The product of the entries along the main diagonal.

Singular matrix

A square matrix whose determinant equals zero, which means it is not invertible.

Effect of a row replacement operation ($R_i \rightarrow R_i + k R_j$) on the determinant

It leaves the determinant unchanged.

Effect of swapping two rows on the determinant

It reverses the sign of the determinant, multiplying it by $-1$.

Effect of multiplying a single row by scalar $k$ on the determinant

It multiplies the determinant by $k$.

Effect of replacing row $R_2$ by $a R_2 + b R_1$ on the determinant

The determinant is multiplied by $a$.

Determinant of a matrix containing a row of all zeros

$$0$$

Determinant of a matrix containing two identical rows

$$0$$

Determinant of a matrix where one row is a scalar multiple of another

$$0$$

Relationship between $\det(A^T)$ and $\det(A)$

$$\det(A^T) = \det(A)$$

Determinant of a product of square matrices ($\det(AB)$)

$$\det(AB) = \det(A) \det(B)$$

Determinant of a positive integer matrix power ($\det(A^k)$)

$$\det(A^k) = (\det(A))^k$$

Determinant of an invertible matrix's inverse ($\det(A^{-1})$)

$$\det(A^{-1}) = \frac{1}{\det(A)}$$

Determinant of a scaled $n \times n$ matrix ($\det(kA)$)

$$\det(kA) = k^n \det(A)$$

Determinant of the $n \times n$ identity matrix ($I_n$)

$$\det(I_n) = 1$$

Invertibility test using determinants

A square matrix $A$ is invertible if and only if $\det(A) \neq 0$.

Row reduction method to compute a determinant

Reduce the matrix to triangular form, multiply diagonal entries, and adjust for any row swaps or row scalings.

Cramer's rule formula for solving $Ax = b$

$$x_i = \frac{\det(A_i)}{\det(A)}$$

Construction of matrix $A_i$ in Cramer's rule

The matrix formed by replacing the $i$-th column of $A$ with the constant vector $b$.

Conditions required to use Cramer's rule

The coefficient matrix $A$ must be square and invertible ($\det(A) \neq 0$).

Geometric area scaling factor of a $2\text{D}$ transformation $T(x) = Ax$

$$|\det(A)|$$

Geometric volume scaling factor of a $3\text{D}$ transformation $T(x) = Ax$

$$|\det(A)|$$

Geometric interpretation of the sign of $\det(A)$

A positive determinant preserves orientation, whereas a negative determinant reverses orientation.

Standard matrix of a linear transformation

The unique matrix $A$ satisfying $T(x) = Ax$ for all vectors $x$.

Matrix inverse formula using the adjugate

$$A^{-1} = \frac{1}{\det(A)} \operatorname{adj}(A)$$

Adjugate of a matrix ($\operatorname{adj}(A)$)

The transpose of the matrix of cofactors.

Vector space

A set of vectors equipped with addition and scalar multiplication satisfying ten algebraic axioms.

Subspace of a vector space

A subset containing the zero vector that is closed under vector addition and closed under scalar multiplication.

Zero-vector requirement for a subspace

The subset must contain the zero vector of the parent vector space.

Closure under vector addition

For all vectors $u$ and $v$ in the subset, the sum $u + v$ must also lie in the subset.

Closure under scalar multiplication

For every vector $u$ in the subset and every real scalar $c$, the product $cu$ must lie in the subset.

Role of a counterexample in testing subspaces

A single instance demonstrating the failure of any subspace axiom proves that the set is not a subspace.

Linear combination

A sum of scalar multiples of vectors: $c_1 v_1 + c_2 v_2 + \dots + c_k v_k$.

Span of a set of vectors

The set of all possible linear combinations formed from the given vectors.

Span-subspace theorem

The span of any set of vectors in a vector space $V$ is automatically a subspace of $V$.

Polynomial space $\mathbb{P}_n$

The vector space of all polynomials with real coefficients of degree at most $n$, including the zero polynomial.

Dimension and standard basis of $\mathbb{P}_n$

The dimension is $n + 1$, and the standard basis is $\{1, t, t^2, \dots, t^n\}$.

Matrix space $M_{m \times n}$

The vector space of all $m \times n$ matrices with real entries.

Vector space role of scalar multiplication versus polynomial multiplication in $\mathbb{P}_n$

Scalar multiplication only scales vectors by real numbers; vector spaces do not define multiplying two polynomials together.

Reason a set of vectors with all nonnegative coordinates fails to be a subspace

Multiplying a nonzero nonnegative vector by a negative scalar produces negative entries, violating closure under scalar multiplication.

Null space of a matrix ($\operatorname{Nul}(A)$)

The set of all solutions $x$ to the homogeneous linear equation $Ax = 0$.

Column space of a matrix ($\operatorname{Col}(A)$)

The span of the original column vectors of $A$.

Row space of a matrix ($\operatorname{Row}(A)$)

The span of the row vectors of $A$.

Condition for a vector $x$ to belong to $\operatorname{Nul}(A)$

$$Ax = 0$$

Condition for a vector $b$ to belong to $\operatorname{Col}(A)$

The linear system $Ax = b$ must be consistent.

Procedure to find a basis for $\operatorname{Nul}(A)$

Solve $Ax = 0$ in parametric vector form; the vectors associated with the free variables form the basis.

Procedure to find a basis for $\operatorname{Col}(A)$

Identify the pivot column positions from an echelon form, and choose the corresponding columns from the original matrix $A$.

Procedure to find a basis for $\operatorname{Row}(A)$

Reduce $A$ to an echelon form; the nonzero rows of that echelon form matrix form the basis.

Free variable

A variable corresponding to a column that does not contain a pivot position in echelon form.

Pivot column

A column in a matrix that contains a leading entry (pivot) when reduced to echelon form.

Rank of a matrix ($\operatorname{rank}(A)$)

The dimension of the column space (or row space), equal to the number of pivot positions.

Nullity of a matrix ($\operatorname{nullity}(A)$)

The dimension of the null space, equal to the number of free variables.

Rank-Nullity Theorem

For any matrix $A$ with $n$ columns: $\operatorname{rank}(A) + \operatorname{nullity}(A) = n$.

Linear transformation

A mapping $T$ satisfying $T(u + v) = T(u) + T(v)$ and $T(cu) = c T(u)$ for all vectors $u, v$ and scalars $c$.

Kernel of a linear transformation ($\ker(T)$)

The set of all vectors $v$ in the domain that map to zero: $T(v) = 0$.

Range (image) of a linear transformation

The set of all image vectors $T(v)$ in the codomain; for $T(x) = Ax$, it equals $\operatorname{Col}(A)$.

Transpose of a matrix ($A^T$)

The matrix formed by interchanging rows and columns, sending entry $(i, j)$ to $(j, i)$.

Symmetric matrix

A square matrix that equals its transpose ($A = A^T$), satisfying $A - A^T = 0$.

Linearly independent set

A set of vectors where $c_1 v_1 + \dots + c_k v_k = 0$ has only the trivial solution $c_1 = \dots = c_k = 0$.

Linearly dependent set

A set of vectors where there exists a nontrivial solution (not all coefficients zero) to $c_1 v_1 + \dots + c_k v_k = 0$.

Nontrivial solution

A solution to a homogeneous system where at least one variable is nonzero.

Independence criterion for two nonzero vectors

Two nonzero vectors are linearly independent if and only if neither vector is a scalar multiple of the other.

Basis for a vector space

A set of vectors that is both linearly independent and spans the vector space.

Column pivot test for linear independence

Place the vectors as the columns of a matrix; they are independent if and only if every column contains a pivot.

Row pivot test for spanning $\mathbb{R}^m$

Place the vectors as columns of an $m \times n$ matrix; they span $\mathbb{R}^m$ if and only if every row contains a pivot.

Determinant basis test for $n$ vectors in $\mathbb{R}^n$

The $n$ vectors form a basis for $\mathbb{R}^n$ if and only if the matrix formed by their columns satisfies $\det(A) \neq 0$.

Independence limitation of more than $n$ vectors in $\mathbb{R}^n$

Any set containing more than $n$ vectors in $\mathbb{R}^n$ is guaranteed to be linearly dependent.

Spanning limitation of fewer than $n$ vectors in $\mathbb{R}^n$

Any set containing fewer than $n$ vectors cannot span $\mathbb{R}^n$.

Basis for a span of independent vectors

Any linearly independent set of vectors forms a basis for the subspace that it spans.

Dimension of a vector space ($\dim(V)$)

The number of vectors in any basis for the vector space.

Spanning Set Theorem (removing vectors)

If a vector in a spanning set is a linear combination of the remaining vectors, it can be removed without changing the span.

Basis Extension Theorem

Any linearly independent set of vectors in a finite-dimensional space can be extended to form a complete basis.

Coordinate vector ($[x]_\mathcal{B}$)

The vector of weights needed to express $x$ as a linear combination of the basis vectors in $\mathcal{B}$.

Form of $[x]_\mathcal{B}$ given $x = c_1 b_1 + \dots + c_k b_k$

$$[x]_\mathcal{B} = (c_1, c_2, \dots, c_k)^T$$

Dimension of a coordinate vector $[x]_\mathcal{B}$

The number of entries equals the number of vectors in basis $\mathcal{B}$, regardless of the ambient dimension.

Standard basis of $\mathbb{R}^3$

$$\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$$

Coordinate vector of $a + bt + ct^2$ relative to standard basis $\{1, t, t^2\}$

$$(a, b, c)^T$$

Change-of-coordinates matrix ($P_\mathcal{B}$)

The matrix whose columns are the vectors of basis $\mathcal{B}$.

Formula converting coordinate vector $[x]_\mathcal{B}$ to vector $x$

$$x = P_\mathcal{B} [x]_\mathcal{B}$$

Formula converting vector $x$ to coordinate vector $[x]_\mathcal{B}$

$$[x]_\mathcal{B} = P_\mathcal{B}^{-1} x$$

Row reduction setup to compute $[x]_\mathcal{B}$

Row-reduce the augmented matrix formed by placing the basis vectors on the left and vector $x$ on the right.

Unique Representation Theorem

Each vector in a vector space with basis $\mathcal{B}$ can be written in uniquely one way as a linear combination of the basis vectors.

Vector space isomorphism

A bijective linear transformation between two vector spaces that preserves vector addition and scalar multiplication.

Isomorphism of $\mathbb{P}_n$ with Euclidean space

$\mathbb{P}_n$ is isomorphic to $\mathbb{R}^{n+1}$ via the coordinate vector mapping relative to the standard basis.

Change of basis

The process of converting the coordinate representation of a vector from one basis to another basis.