Linear Algebra Part 2 : VECTOR SPACES ( BASIS & DIMENSION)

Explore the definition of a basis as a linearly independent generating set in vector spaces, and demonstrate with R^2, R^3, and R^n using standard basis vectors.

Show the given vectors form a basis for R3 by proving linear independence and that every R3 vector is a linear combination of them, via determinant.

Identify the y z plane as a subspace of R3 spanned by the vectors (0,1,0) and (0,0,1). Show the xy plane is the subspace spanned by (1,0,0) and (0,1,0).

Explain how to find bases for 2x2 real, symmetric, and skew-symmetric matrices by expressing any matrix as a linear combination of specific basis matrices.

Explore bases for 2x2 Hermitian and skew Hermitian matrices over the real field, derive generators, and prove linear independence to establish explicit bases.

Identify the basis for the vector space of all two by three real matrices by expressing any element as a linear combination of six standard basis matrices, proving independence.

Show that the vector space of real symmetric matrices is spanned by three given matrices, they are linearly independent, and a nonzero determinant confirms a trivial solution, forming a basis.

The lecture shows that any m by n matrix A equals sum a_ij E_ij, and that 1, x, x^2, ..., x^n form a standard basis for polynomials degree ≤ n.

Prove that 1, 1+x, and 1+x+x^2 form a basis for real polynomials of degree at most two by verifying independence and generating every polynomial in V.

Prove that the polynomials {1, 1+x, 1+x+x^2, ..., 1+x+...+x^n} form a basis for the real polynomials of degree at most n by establishing linear independence and spanning.

Demonstrate that the polynomials 1, 2−x, 3+x^2, 4−x^3 form a basis for real polynomials degree ≤3 by proving linear independence and generating all vectors in V.

Demonstrate that the columns of an invertible n by n matrix form a basis for the space of f n by 1 column matrices, by proving linear independence and spanning.

Explore the concept of a basis for a vector space v: verify that vectors v1,...,vn generate v and express every element uniquely, proving linear independence and completeness.

Show why any n+1 vectors in a vector space generated by n vectors are linearly dependent, by forming a homogeneous system and identifying a nontrivial solution.

Prove that any independent set in a vector space generated by a finite set contains at most n vectors, by showing that any m>n is linearly dependent.

Demonstrate that any two bases of a finitely generated vector space over F have the same number of elements.

Defines the dimension of a finite vector space as the number of vectors in any basis, with examples like R3 is 3-dimensional and polynomials degree ≤ n have dimension n+1.

Demonstrate that in a finite dimensional vector space V with dim V = n, no spanning set contains fewer than n vectors. Use a basis to derive a contradiction.

In a finite dimensional vector space V with dim V = n, a generating set of n elements is a basis, and no smaller set can span V.

Explore the existence theorem in finite dimensional vector spaces: prove that a linearly independent n-vector set is a basis for V and that any larger set is linearly dependent.

Show that a subspace W of a finite dimensional vector space V satisfies dim W ≤ dim V by extending a nonzero vector in W to a basis.

Demonstrate that any proper subspace W of a finite-dimensional V has dim W < dim V by extending a basis of W to a basis of V.

Prove that for finite dimensional subspaces W1 and W2 of V, dim(W1+W2) = dim W1 + dim W2 − dim(W1∩W2), establishing finite dimensionality and a basis for the sum.

Prove that if the intersection of two finite-dimensional subspaces is zero, then the dimension of their sum equals the sum of their dimensions, using bases and linear independence.

Demonstrate basis construction for polynomials degree ≤ n using 1+x, 1+x^2, ..., 1+x^n, showing linear independence and dimension n+1; apply subspace intersection and sum dimensions to R3 examples.

Explore solved examples of vector subspaces in R3. Apply the dimension formula dim(W1+W2) = dim W1 + dim W2 - dim(W1 ∩ W2) to show the intersection is nonzero.

Extend w1's basis to a full basis of V to form w2, show V = w1 + w2 and w1 ∩ w2 = {0}, proving the direct sum of subspaces.

Show that R3 equals w1 plus w2 and w1 ∩ w2 = {0}, using dim w1 = 1 and dim w2 = 2 with bases.

Explore the definition of a coordinate vector relative to an ordered basis by expressing vectors as linear combinations and extracting the coordinate scalars; solve examples in vectors, matrices, and polynomials.