V3(R) represents the set of vectors in 3D space. What kind of geometrical objects are represented by the various subspaces of V3(R)?

i.e A 1D subspace S with basis { (0, 1, 0)Transpose} represents the set of vectors parallel to the y-axis, so the set of points with position vectors in s is the y-axis itself.

You need only consider subspaces of dimension less than 3.

Solution Preview

A line passing through origin represents the 1-dimensional subspace of V3.
Here are some examples:
S1 = <(1,0,0)^T> is the x-axis
S2 = <(0,1,0)^T> is the y-axis
S3 = <(0,0,1)^T> is the z-axis
Now we ...

Solution Summary

Vector Spaces, Subspaces and Dimension are discussed. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Questions regarding vector spaces. 9. Let u, x, y, z be elements in a vector space V where. ... a basis for the space spanned by the following vectors (see attached ...

... What is the dimension of space spanned by these vectors? 2. Find the orthogonal projection of the vector u = (-2, 2, 3) onto the subspace spanned by vectors v1 ...

... 9125;⎢⎥⎣⎦ Vector space, Basis, Dimensions... W be the subspace of 4R generated by the vectors ()()1,2 ... of W to a basis of the whole space 4R ...

... be two subspaces of a finite dimensional vector space V such ... U2 = V. Prove that there is a subspace W of ... consider the span of the vectors that have been added ...

... to zero v) The 3×3 matrices A such that vector [1; 2 ... 4. Find a basis for the spaces and determine its dimension... i) The space of all 2X2 matrices A such that A[1 ...

... a division ring.) Let V be an n-dimensional vector space over K ... A is clearly Artinian since it has finite dimension over K ... 1. Let W be a subspace of V. Then LW ...

... the square root of the sum of the square of each components, just like the 2-norm of a vector. ... A linear space E admits a basis of orthonormal vectors v1 ...

... Let V be an n-dimensional vector space over K and let A ... ring is clearly Artenian since it has finite dimension over K ... either (0) or A. Let W be a subspace of V ...

... f is the matrix A such that for every vector v in ... 4. (a) We are given an inner product space V and a map We ... g is an isometry if for every pair of vectors u and ...