Artin Problems: Section 3.3

Problems from Chapter 3: Vector Spaces, Section 3: Vector Spaces

Problem 3.3.1

(a) Prove that the scalar product of a vector with the zero element of the scalar field F is the zero vector.

Since scalar multiplication is linear in vector fields. 0 \cdot (x_1,...x_n) = (0\cdot x_1,...,0\cdot x_n) = (0,...,0)

(b) Prove that if w is in the vector space W then -w is in W also.

Since w \in W the span of w is a subspace of W.

If -w \not \in span(w) then there exists no linear combination such that a_1w+a_2(-w) = 0.

This is false for a_1,a_2 = 1 therefore -w \in span(w) \subset W.

Problem 3.3.2

Which of the following subsets is a subspace of the vector space F^{n \times n} of n \times n matrices with coefficients in F?

(a) symmetric matrices, i.e. \{A: A = A^t\}.

Can change condition that A = A^T to equivalent condition that a_{ij} = a_{ji} \forall i,j.

Addition preserves space membership. A+B = C gives matrix C such that c_{ij} = a_{ij}+b_{ij} Obviously swapping i,j gives c_{ij} = c_{ji}, therefore C symmetric.

scalar multiplication preserves symmetry and zero matrix is symmetric.

Therefore symmetric matrices form a vector subspace.

 (b) invertible matrices.

Not a vector subspace as 0 matrix is not invertible.

(c) upper triangular matrices.

Obviously if $latex A,B$ upper triangular then A+B is upper-triangular as addition of matrices is elementwise.

cA is also upper triangular as multiplication by a scalar preserves zeroes.

The zero matrix is also an upper triangular matrix, as the definition requires only that if i >j then c_{ij} = 0 and not that if j \leq i then c_{ij} \neq 0

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