Vectors Generating R^3 Linear Algebra Problem

Here are the solutions to the problems:

Problem 2:
In , let . Is a subspace of ?

To determine if is a subspace of , we need to check three conditions:

Let be the vector space of polynomials of degree at most 4 with real coefficients.

Since all three conditions are met, is a subspace of .

Answer for Problem 2: Yes, is a subspace of .

Problem 3:
Let . Is a vector subspace of ?

To determine if is a subspace of , we need to check the three conditions:

Since is not closed under addition, it cannot be a subspace of . (We don’t even need to check scalar multiplication, but for completeness, let’s check it).

Answer for Problem 3: No, is not a vector subspace of .

Problem 4:
Determine if the set generates .

A set of vectors generates if any vector can be written as a linear combination of the vectors in the set . That is, we need to find scalars such that:

This gives us a system of linear equations:

Now we can solve this system:
From the third equation, .
Substitute into the first equation: .
Substitute into the second equation: .

Since we can always find unique values for for any given , the set generates .

Alternatively, we can check if the vectors are linearly independent. If they are, and there are 3 vectors in , then they form a basis and thus generate .
We can form a matrix with these vectors as rows or columns and calculate its determinant.
Let the matrix be:

The determinant of is:

Since the determinant is non-zero ( ), the vectors are linearly independent.
Since there are 3 linearly independent vectors in (which has dimension 3), they form a basis for and therefore generate .

Answer for Problem 4: Yes, the set generates .