For question 7 of your lab assignment:

We are tasked with showing that the polynomial $S = 3x^3 + 2x^2$ is not a linear combination of $P$ and $Q$, where:

• $P = 2x^3 + x^2 + 3x + 4$
• $Q = x^3 + x^2 - 5x + 3$

Approach:

A polynomial is a linear combination of other polynomials if it can be expressed as $S = aP + bQ$, where $a$ and $b$ are real numbers.

This means we need to check if there exist values $a$ and $b$ such that: $S = a(2x^3 + x^2 + 3x + 4) + b(x^3 + x^2 - 5x + 3)$

Expanding the right-hand side:

$a(2x^3 + x^2 + 3x + 4) + b(x^3 + x^2 - 5x + 3) = (2a + b)x^3 + (a + b)x^2 + (3a - 5b)x + (4a + 3b)$

Now, equate this to $S = 3x^3 + 2x^2$, which gives us:

$(2a + b)x^3 + (a + b)x^2 + (3a - 5b)x + (4a + 3b) = 3x^3 + 2x^2$

Compare the coefficients:

For this equality to hold, the coefficients of corresponding powers of $x$ must be equal. This gives the following system of equations:

1. For $x^3$: $2a + b = 3$
2. For $x^2$: $a + b = 2$
3. For $x^1$: $3a - 5b = 0$
4. For the constant term: $4a + 3b = 0$

Solving the system:

• From equation (2): $a = 2 - b$
• Substitute $a = 2 - b$ into equation (1): $2(2 - b) + b = 3 \implies 4 - 2b + b = 3 \implies -b = -1 \implies b = 1$
• Substitute $b = 1$ into $a = 2 - b$: $a = 2 - 1 = 1$

Now, check the other two equations:

• Substitute $a = 1$ and $b = 1$ into equation (3): $3(1) - 5(1) = 0 \implies 3 - 5 = -2 \quad \text{(not satisfied)}$ Since this equation is not satisfied, there are no values of $a$ and $b$ that satisfy all four equations.

Conclusion:

Therefore, $S = 3x^3 + 2x^2$ is not a linear combination of $P$ and $Q$.

Let me know if you'd like a more detailed explanation or have any questions! Here are 5 related questions:

1. What does it mean for polynomials to be linearly independent?
2. How can you test if two polynomials form a basis for a vector space?
3. Can a polynomial of degree 4 be a linear combination of polynomials of lower degrees?
4. How do the coefficients of a polynomial relate to its representation in matrix form?
5. What are some practical applications of polynomials in linear algebra?

Tip: When checking if a polynomial is a linear combination, focus on comparing the coefficients of corresponding powers of $x$.