MATH2051 - Linear Algebra II Flashcards

Vectors and Operations

• A vector in the plane is represented as a 2x1 matrix x=[x y], with x and y as real number components.
• The sum of two vectors u = [u1 u2] and v = [v1 v2] results in u + v = [u1 + v1, u2 + v2].
• Scalar multiplication of vector u = [u1 u2] by scalar c yields cu = [cu1, cu2], altering the direction if c < 0.

Properties of Vector Spaces

• Vector addition is commutative and associative; adding the zero vector returns the original vector.
• Each vector u has an additive inverse denoted by -u, satisfying u + (-u) = 0.
• Scalar multiplication distributes over vector addition and respects scalar addition and multiplication rules.

Subspaces

• A non-empty subset W of a vector space V is a subspace if it is closed under vector addition and scalar multiplication.

Linear Combinations and Spanning

• A vector v in vector space V is a linear combination of vectors v1, v2,..., vk if v = a1v1 + a2v2 + ... + akvk for real coefficients a1, a2,..., ak.
• The span of a set of vectors S consists of all possible linear combinations of vectors in S.
• Span S is a subspace of V.

Linear Independence and Basis

• Vectors v1, v2,..., vk are linearly dependent if a non-trivial linear combination equals zero, otherwise they are independent.
• A set of vectors forms a basis for vector space V if they span V and are linearly independent.
• The determinant of a matrix whose columns are vectors indicates linear independence; specifically, non-zero determinant implies independence.

Theorems on Linear Combinations

• Linear combinations can reveal independence; if any vector in a set is dependent on the preceding vectors, then the entire set is dependent.
• Any basis S of vector space V guarantees that every vector in V can be uniquely expressed as a linear combination of vectors from S.

Dimension

• The dimension of a vector space V is defined as the count of vectors in any basis for V.

Standard Basis

• For R^3, the standard basis is defined as {[1, 0, 0], [0, 1, 0], [0, 0, 1]}.
• For polynomial space Pn, the standard basis consists of {t^n, t^(n-1), ..., t, 1}.