Algebra Practice Exercises Flashcards

A comprehensive set of vocabulary flashcards covering sets, mappings, complex numbers, matrices, vector spaces, and linear transformations from the MI1036 Advanced Program Algebra course.

Intersection of Sets ($A \cap B$)

The set of elements that belong to both set $A$ and set $B$.

Union of Sets ($A \cup B$)

The set of elements that belong to at least one of the sets $A$ or $B$.

Difference of Sets ($A \setminus C$)

The set containing elements that are in set $A$ but not in set $C$.

Cartesian Product ($A \times B$)

The set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

Injective Map

A mapping $f: X \rightarrow Y$ where different elements in the domain map to different elements in the codomain (one-to-one).

Surjective Map

A mapping $f: X \rightarrow Y$ where every element in the codomain $Y$ has at least one pre-image in the domain $X$ (onto).

Bijective Map

A mapping that is both injective and surjective, meaning every element in the domain corresponds to exactly one element in the codomain and vice-versa.

Associative Operation

A binary operation $*$ such that $(a * b) * c = a * (b * c)$ for all elements in the set.

Commutative Operation

A binary operation $*$ such that $a * b = b * a$ for all elements in the set.

Group

A set equipped with a binary operation that is associative, contains an identity element, and where every element has an inverse.

Modulus Operation ($a \pmod n$)

The operation that finds the remainder of the integer $a$ when divided by the positive integer $n$.

Canonical Form of a Complex Number

The representation of a complex number in the form $z = a + bi$, where $a$ is the real part and $b$ is the imaginary part.

Polar Form of a Complex Number

The representation of a complex number as $z = r(\cos(\theta) + i\sin(\theta))$ where $r$ is the modulus and $\theta$ is the argument.

Matrix Transpose ($A^T$)

An operation that flips a matrix over its diagonal, switching its row and column indices.

Symmetric Matrix

A square matrix $A$ that is equal to its transpose ($A = A^T$).

Skew-symmetric Matrix

A square matrix $A$ whose transpose is equal to its negative ($A^T = -A$).

Invertible Matrix

A square matrix $A$ for which there exists a matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$, where $I$ is the identity matrix.

Determinant ($\det(A)$)

A scalar value calculated from the entries of a square matrix that characterizes certain properties of the linear transformation or the matrix's invertibility.

Vector Space

A collection of objects (vectors) that can be added together and multiplied by scalars while following specific axioms such as associativity and distributivity.

Subspace

A subset of a vector space that is itself a vector space under the original addition and scalar multiplication operations.

Direct Sum ($V_1 \oplus V_2$)

A way of combining two subspaces such that their intersection is only the zero vector and every vector in the sum space has a unique representation as a sum of one vector from each subspace.

Linear Combination

An expression constructed from a set of vectors by multiplying each vector by a constant (scalar) and adding the results.

Linear Independence

A property of a set of vectors where no vector in the set can be represented as a linear combination of the other vectors.

Span

The set of all possible linear combinations of a given set of vectors.

Basis

A set of linearly independent vectors that spans a vector space, serving as a coordinate system.

Dimension ($\dim(V)$)

The number of vectors contained in any basis of a vector space $V$.

Coordinate Vector ($[x]_S$)

The column vector of coefficients used to represent a vector $x$ as a linear combination of the basis vectors in set $S$.

Rank of a Matrix

The maximum number of linearly independent rows or columns in a matrix.

Linear Transformation

A mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.

Kernel ($\text{Ker}T$)

The set of all vectors in the domain that map to the zero vector of the codomain under the linear transformation $T$.

Image ($\text{Im}T$)

The set of all vectors in the codomain that are the result of the linear transformation $T$ acting on the domain.

Eigenvalue ($\lambda$)

A scalar associated with a linear transformation such that there exists a non-zero vector $v$ (eigenvector) where $T(v) = \lambda v$.

Eigenvector

A non-zero vector that changes only by a scalar factor when a specific linear transformation is applied to it.

Trace ($\text{tr}(A)$)

The sum of the elements on the main diagonal of a square matrix.

Inner Product ($\langle u, v \rangle$)

A rule that assigns a scalar to each pair of vectors in a vector space, satisfying properties like positivity, symmetry, and linearity.

Orthogonal Vectors

Two vectors $u$ and $v$ are orthogonal if their inner product is zero ($\langle u, v \rangle = 0$).

Gram-Schmidt Process

An algorithm used to convert a set of linearly independent vectors into an orthonormal basis for the same subspace.

Orthogonal Projection

The unique vector in a subspace that is "closest" to a given vector outside that subspace, determined by orthogonality.

Least Square Solution

The vector $X$ that minimizes the norm of the error function $f(X) = \|AX - B\|$ for a linear system that may not have an exact solution.

Quadratic Form

A homogeneous polynomial of degree two in a number of variables, often represented as $x^T Ax$ for a symmetric matrix $A$.

Positive Definite Quadratic Form

A quadratic form $\omega(x)$ where $\omega(x) > 0$ for every non-zero vector $x$.