Please provide solutions to these two questions (attached).
Please show how the subspace satisfies both addition & scalar multiplication!

In each of the following exercises 8-17, we will denote by S the set of all vectors x = (x1, x2, x3) E R3 whose coordinates satisfy the given condition. In each case determine whether the given set S is a linear subspace of R3 and if so, compute its dimension.

16. x1 = x2 = x3

15. x2 = 2x1 and x3 = 3x1

Partial solution to 15:

{x = (x1, x2, x3) E R3: x2 = 2x1, x3 = 3x1}

So x = (x1, 2x1, 3x1)
= x1(1, 2, 3)

So that shows that the subspace is closed under scalar multiplication. And it obviously contains the zero vector: (0, 0, 0). What about addition though?---

... The problems are from Linear Algebra. ... Solutions are given in the attached file. Question (1) Find a basis and dimension of the subspace W of R4 generated by ...

... to (17), the subspace W will be spanned by the basis B1: ( 19) and its dimension is 2 as it was assumed before. (The vectors of the basis are linear independent ...

... Let E be a linear subspace of n . Show that E is a closed subspace of n ... A linear space E admits a basis of orthonormal vectors v1 ... vm , here the dimension of E ...

Linear Algebra: Vectors and Least Squares Problems. ... What is the dimension of space spanned by these vectors? ... vector u = (-2, 2, 3) onto the subspace spanned by ...

... Find a basis for the spaces and determine its dimension. ... vi) It is not a subspace, since the sum of ... is equivalent to the following system of linear equations. ...

Linear algebra. ... vector space V such that U1+U2 = V. Prove that there is a subspace W of U1 such that W (+) U2 = V. [as in proof of dimension thm, extend to a ...

... A is clearly Artinian since it has finite dimension over K ... function of f | f I } is a subspace of V ... H is the annihilator of W, and so any linear function V ...

... ring is clearly Artenian since it has finite dimension over K ... either (0) or A. Let W be a subspace of V ... elements of the ring A with the linear transformations V ...

... Let W be the subspace of 4R generated by the vectors ()()1,2,5,3,2,3 ... 8722;− and ()3,8,3, a. Find a basis and the dimension of W ... System of linear equation 21 ...