2010年8月23日 · Orthogonal Projection Onto a Subspace is used in a variety of fields, such as linear algebra, statistics, and machine learning. It is used to find the closest approximation of a vector to a subspace and is also an essential tool in regression analysis. Can Orthogonal Projection Onto a Subspace be applied to higher dimensions?

2011年6月10日 · In summary, a closed linear subspace in a Hilbert space is a subspace that remains closed under scalar multiplication and vector addition, and all converging sequences within the subspace will converge to points within the subspace. It is like a "prison" where points cannot escape, even in the limit.

2016年1月23日 · How can one know how small a subspace is? initially I thought it was determined from the number of elements in the subspace, but there infinite number of elements. Can also someone please give an example by giving two subspaces and show the ways to compare which one is smaller than which?

2012年10月31日 · In summary, the subspace topology generated by taking the Rationals as a subset of the Reals can be visualized as infinite sets of rational numbers within open intervals, since each open set in ℝ is an open interval and the open intervals form a basis for ℝ. However, in the subspace topology, open singleton sets are not possible.

2011年4月27日 · A subspace is different from a vector space in that it is a subset and not all subspaces are vector spaces. A subspace can have a dimension that is equal to or less than the dimension of the original vector space, determined by the number of …

2006年8月25日 · That said, originally, I was a little surprised by the question. It is common to think of R^2 as being a subset of R^3 using the obvious isomorphism to a subspace of R^3: (a, b)-> (a, b, 0). Strictly speaking, it is not R^2 that is a subspace of R^3, it …

2013年10月15日 · I want to know why this subset W is a subspace of R3. W is defined as: | x+2y+3z | | 4x+5y+6z | | 7x+8y+9z | I know the possible subspaces of R3 are the origin itself, lines through the origin, and planes through the origin. Would W be a subspace of R3 simply because there would be...

2012年11月22日 · It is. But it is not a subspace. I suspect that was a typo. Yes, this is sufficient. To be a subspace, the subset must satisfy a number of properties. If it fails to satisfy anyone of them it is not a subspace.

2014年5月21日 · As CAF123 said, a subset isn't necessarily a subspace. A subspace of V is a subset of V that's also a vector space (with the addition and scalar multiplication operations inherited in the obvious way from the vector space). So the given subset is a subspace if and only if it satisfies the vector space axioms.

2015年3月15日 · The largest subspace in the intersection of U, V, W would be a subspace which has all possible 4x4, symmetric, upper triangular matrices. Otherwise--the smallest subspace of any space is the 0 space. The largest space question seems clear either way.