Intro to subspaces - example clarificatio

[Glaussesness]

Hi,
My textbook gives some examples relating to subspaces but I am having trouble intuiting them.
Could someone please help me understand the five points they are attempting to convey here (see screenshot).
Again, I am aware that these are rather beginner but I would like to have an extremely solid foundation before moving forward - walking me through the finer points would be amazing.
Your patience with a fool is appreciated

 

[pka]

My textbook gives some examples relating to subspaces but I am having trouble intuiting them. Could someone please help me understand the five points they are attempting to convey here (see screenshot).
Again, I am aware that these are rather beginner but I would like to have an extremely solid foundation before moving forward - walking me through the finer points would be amazing.

Glaussesness, on this forum you must post some work so that we know how to help.
To start. tell us how one shows that a set is a subspace of a space.
Then say how one would start doing part a).

 

[Steven G]

Part(a): A subspace must be closed under addition and closed under scalar multiplication.

You can pick any values for x₁, x₂ and x₄ in F, but x₃ must equal 5x₄ + b.

Now take any two vectors in F and add them. Under what conditions do you get back another vector in F. Then do the same for scalar multiplication. What are the conditions on b?

 

[HallsofIvy]

In the first problem any vector is of the form \(\displaystyle \left(x_1, x_2, 5x_4+ b, x_4\right)\) so two examples are (1, 3, 10+ b, 2) and (7, 3, -6+ b, -3). In a vector space, the sum of two vectors is a vector. The sum of those is (8, 6, 4+ 2b, -1). Is that of the form \(\displaystyle \left(x_1, x_2, 5x_4+ b, x_4\right)\)?

Also, the product of a scalar and a vector must be a vector.
2 times (1, 3, 10+ b, 2) is (2, 6, 20+ 2b, 4). Is that of the form
\(\displaystyle \left(x_1, x_2, 5x_4+ b, x_4\right)\)?