The image contains two parts of a linear algebra problem regarding polynomials and their coordinates with respect to a basis. Here's how to break down and solve the problem:

1. Basis for Cubic Polynomials:

The polynomials $1 + x$, $1 + x^2$, $1 - x + x^2$, and $x + x^3$ are given to form a basis for the vector space of all cubic polynomials. A cubic polynomial has the general form:

$p(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3$

Since there are four polynomials, and they are independent, they form a basis for this vector space.

2. Coordinate of $1 + x$ with respect to the basis:

The task is to find the coordinate of the polynomial $1 + x$ with respect to the given basis. Let this coordinate be denoted as:

$[1 + x] = \begin{bmatrix} c_1 \\ c_2 \\ c_3 \\ c_4 \end{bmatrix}$

where $c_1, c_2, c_3, c_4$ are the scalars that correspond to the basis vectors. We need to solve for these scalars such that:

$1 + x = c_1(1 + x) + c_2(1 + x^2) + c_3(1 - x + x^2) + c_4(x + x^3)$

Expanding both sides:

$1 + x = c_1(1 + x) + c_2(1 + x^2) + c_3(1 - x + x^2) + c_4(x + x^3)$

Step-by-step expansion:

Expand each term:

• $c_1(1 + x) = c_1 + c_1 x$
• $c_2(1 + x^2) = c_2 + c_2 x^2$
• $c_3(1 - x + x^2) = c_3 - c_3 x + c_3 x^2$
• $c_4(x + x^3) = c_4 x + c_4 x^3$

Summing all terms:

$1 + x = (c_1 + c_2 + c_3) + (c_1 - c_3 + c_4)x + (c_2 + c_3)x^2 + c_4 x^3$

Now, equate coefficients of $1$, $x$, $x^2$, and $x^3$ from both sides:

• Constant term: $1 = c_1 + c_2 + c_3$
• $x$-term: $1 = c_1 - c_3 + c_4$
• $x^2$-term: $0 = c_2 + c_3$
• $x^3$-term: $0 = c_4$

Solving the system of equations:

We now have the system of equations:

1. $c_1 + c_2 + c_3 = 1$
2. $c_1 - c_3 + c_4 = 1$
3. $c_2 + c_3 = 0$
4. $c_4 = 0$

From equation (4), we know $c_4 = 0$. Substitute this into equation (2):

$c_1 - c_3 = 1$

From equation (3), $c_2 = -c_3$. Substitute this into equation (1):

$c_1 - c_3 + c_3 = 1 \quad \Rightarrow \quad c_1 = 1$

Thus, $c_1 = 1$. Now, using equation (3), $c_2 = -c_3$, and substituting into equation (1):

$1 + (-c_3) + c_3 = 1$

This holds true for any $c_3$, so $c_2 = 0$ and $c_3 = 0$.

Conclusion:

The coordinate vector of $1 + x$ with respect to the basis is:

$[1 + x] = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}$

Would you like more details or have any questions? Here are some related questions:

1. How do you verify if a set of polynomials forms a basis for a given space?
2. What is the significance of the dimension of a polynomial vector space?
3. Can a polynomial of degree less than 3 be expressed as a linear combination of cubic polynomials?
4. How do you find the coordinates of a polynomial in a different basis?
5. Why are polynomial spaces important in functional analysis?

Tip: Always check the independence of your basis vectors when working with polynomial spaces to ensure they span the entire space.