4) Which of the following is a subspace of R^3? Justify your answer. i) W={(x,y,z)|x+y+z=0} ii) W={(x,y,z)|x^2+y^2+z^2=0} iii) W={(x,y,z)|x>0} iv). W={(x,2x,3x) |x is a real number}
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- Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector u=(1,1,1,1) in the form u=v+w, where v is in V and w is orthogonal to every vector in V.Question: Is the subset P(x,y,z) described by 8x-y+2z=0 a subspace of R3? Why or why not? Is my interpretation of subspace correct? Are there any explicit theorems that explain this concept? This is what I came up with: The subset P(x,y,z)described by 8x - y + 2z = 0 is a subspace of R3 because the plane passes through the origin which is used as the positional matrix. Any scalar of x,y, or z will equal 0 and this is an example of a special subspace called the null space. The system is consistent and has infinitely many solutions.
- How would I find whether there is a linear algebra subspace in R^3, inclusion of zero vector, closure under vector addition or scalar multiplication?can we have a polynomial Q(x,y,z) such that the subspace {(x,y,z):Q(x,y,z)=0} is a smooth submanifold diffeomorphic to T^2(torus) in step by step with out using chatgpt.correction: If the solution space to the homogeneous linear system Ax = 0 is a 2-dimensional subspace...
- Let S be the set of all 2 × 2 real matrices given by Find 2 × 2 matrices X, Y, Z such that S = span{X, Y, Z}. Why does this show that S is a subspace of R^2×2?Find a basis for the subspace given by the plane −3x + 2 y + 5z = 0.We denote the subspace spanned by the first two columns of A by U, and the subspace spanned by the last three columns of A by V . It's asked to determine a basis of U ∩ V, Please do it step by step, I got to the point of determining the null space which is (1/3, -1/3, 1, 1, 1)x5, but what happens after that?
- Question involving 3 sub parts, which of these are subspaces?(5.4) (6) Find the projection of the vector v onto the subspace S. S = span 0 1 1 , 1 1 0 v = 3 4 2 projs v = ??Consider the vector subspaces U, V ,W ⊆ Rn such that U ⊆ V ⊆ W. If dimU = 5 and dimW = 7, what can be said about V? -It always contains a set of 5 linearly independent vectors -dimV = 6 -it always contains a set of 7 linearly independent vectors -the zero vector is not contained in it -it contains a non-zero vector