Linear Algebra- Part 1 (Vector Spaces and Subspaces)

Learn that the real-valued function space decomposes into two subspaces: even functions and odd functions, whose intersection is the zero function and whose sum equals the whole space.

Learn to verify subspaces of the vector space of real and complex valued functions using nonemptiness and closure, with examples like derivative equals zero, derivative equals x^2, and zero integral.

This lecture presents solved examples on vector spaces, verifying subspace criteria for various subsets of R^3, including closure under addition and scalar multiplication, and identifying non-subspaces through counterexamples.

Explore solved examples on vector spaces with matrices, proving subspace properties via zero vectors and linear combinations, and examining sets of column vectors and matrix relations in different dimensions.

Show that homogeneous system's solution set is a subspace and that W1 and W2 with intersection {0} yield unique V = V1 + V2 for V in W1 + W2.

Define the direct sum of subspaces in R3 using W1 as the XY plane and W2 as the Z axis, proving R3 = W1 + W2 with intersection {0}.

Explore direct sum of subspaces in linear algebra, including decomposition into symmetric and skew-symmetric matrices, with concrete examples and proofs of intersection zero.

Show that R^3 equals W1 plus W2 with trivial intersection, illustrating a direct sum; also study 2x2 matrices with zero second row or column as subspaces.

Demonstrate that V equals the direct sum of W1 and W2 when V = W1 + W2 and W1 ∩ W2 = {0}, showing unique representations from W1 and W2.

Learn how a subspace of a subspace is itself a subspace. Verify subspace postulates, and explore direct sums, intersections, and sums of subspaces with examples of triangular and diagonal matrices.

Define linear combinations of vectors and the span of a set, show this span forms a subspace, and prove it is the smallest subspace containing the given vectors.

Explore the linear span of a vector set and how it generates the smallest subspace of a vector space, using linear combinations to characterize generating sets and subspace containment.

Show that the linear span L(S) equals the intersection of all subspaces of V containing S, using linear combinations to characterize containment.

Demonstrate that if S is a subset of T, then L(S) is contained in L(T) and prove the reverse to establish equality, with examples generating R^3 from given vectors.

Examine finitely generated vector spaces and why the space of all real polynomials is not finitely generated, and learn linear independence via homogeneous systems and determinants.

Explore when homogeneous systems have non-trivial solutions, and distinguish singular from non-singular matrices using determinants and inverses, with examples of vector independence and dependence.

Explore linear independence and dependence, proving unique representations in a basis and that a zero vector induces dependence, while showing that subsets of independent sets remain independent.

Learn that every subset of a linearly independent set remains linearly independent, and that a linearly dependent set contains a vector that is a linear combination of the others.

Prove a proposition on linear dependence and derive the corollary that none of the vectors is a linear combination of preceding vectors, hence the set is linearly independent.

Learn when vector sets are linearly dependent or independent in vector spaces, including that {v} is dependent iff v=0 and {v1,v2} iff one is a scalar multiple of the other.

Explore coplanar and collinear vectors through exercises, proving when two vectors are linearly independent or dependent, and when three vectors are coplanar via linear combinations.

Identify when a nonzero vector is a linearly independent singleton, two vectors are dependent iff one is a scalar multiple, and discuss extending a generating set while preserving generation.

Examine how to determine linear independence and basis for vector spaces by solving homogeneous systems and checking the determinant of the coefficient matrix, illustrated with a polynomials example.