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Subspaces of R3: Proof or Counterexample • Physics Forums missed last week due to illness so have no clue what this homework is going on about, the question is for each of the following state whether or not it is a subspace of R3, Justify your answer by giving a proof or a counter example in each case, i know I'm ment to attempt the question before i
LINEAR ALGEBRA: Consider 2X2 Matrices - Physics Forums The discussion focuses on identifying which sets of 2x2 matrices qualify as subspaces of R^2x2 based on specific conditions It is concluded that sets (A), (B), and (D) are subspaces, while set (C) is not because the sum of two matrices with a zero determinant can yield a matrix with a non-zero determinant, violating the subspace criteria Set (E) is also dismissed as a subspace due to failing
Subspaces of R2 and R3: Understanding Dimensions of Real Vector Spaces So I'm considering dimensions of real vector spaces I found myself thinking about the following: So for the vector space R2 there are the following possible subspaces: 1 {0} 2 R2 3 All the lines through the origin Then I considered R3 For the vector space R3 there are the
Finding Basis for Subspaces of R4 - Physics Forums The discussion revolves around finding bases for subspaces W and U in R4 and determining their intersections and sums The basis for W is identified as a (1,1,1,0) + b (-1,2,0,1), leading to a homogeneous system of equations For W ∩ U, a comparison of the systems reveals a basis of (-0 5, 4, 1, 1 5) The basis for W + U is proposed as four vectors, but it is later clarified that they are
Find a basis for the space of 2x2 symmetric matrices The discussion highlights that symmetric matrices form a subspace of the larger space of 2x2 matrices, reinforcing the properties of vector spaces over a field
Is a subspace still valid without the zero vector? - Physics Forums No, because the subspace will have negatives of elements, i e , for all v an element of V, (-1)v or -v will be an element For the subspace to be closed under addition (a necessary requirement) v + (-v) = 0 must be an element which implies the zero vector must be in a subspace of vectors