Let's work through both of the problems step by step.

1. Express $\left( \frac{1}{2}x^2 - x \right) \in \text{Span}\{ 1, x, x^2 \}$:

We want to express the polynomial $\frac{1}{2}x^2 - x$ as a linear combination of the vectors in the span $\{1, x, x^2\}$. This means we need to find scalars $a$, $b$, and $c$ such that:

$\frac{1}{2}x^2 - x = a(1) + b(x) + c(x^2)$

Now, equate the terms:

$\frac{1}{2}x^2 - x = a + bx + cx^2$

Now, match the coefficients of like powers of $x$:

• For $x^2$, we have: $c = \frac{1}{2}$
• For $x$, we have: $b = -1$
• For the constant term, we have: $a = 0$

Thus, the linear combination is:

$\frac{1}{2}x^2 - x = 0 \cdot 1 + (-1) \cdot x + \frac{1}{2} \cdot x^2$

2. Express $(1, 2, 3) \in \text{Span}\{(1, 1, 1), (1, 1, 0), (1, 0, 0)\}$:

We want to express the vector $(1, 2, 3)$ as a linear combination of the vectors $(1, 1, 1)$, $(1, 1, 0)$, and $(1, 0, 0)$. This means we need to find scalars $p$, $q$, and $r$ such that:

$(1, 2, 3) = p(1, 1, 1) + q(1, 1, 0) + r(1, 0, 0)$

Now, expand the right-hand side:

= (p + q + r, p + q, p)$$ Now, equate the components: - For the first component: $$p + q + r = 1$$ - For the second component: $$p + q = 2$$ - For the third component: $$p = 3$$ From the third equation, $$p = 3$$. Substitute this into the second equation: $$3 + q = 2 \implies q = -1$$ Now substitute $$p = 3$$ and $$q = -1$$ into the first equation: $$3 - 1 + r = 1 \implies r = -1$$ Thus, the linear combination is: $$(1, 2, 3) = 3(1, 1, 1) + (-1)(1, 1, 0) + (-1)(1, 0, 0)$$ ### Summary of the linear combinations: 1. $$\frac{1}{2}x^2 - x = 0 \cdot 1 + (-1) \cdot x + \frac{1}{2} \cdot x^2$$ 2. $$(1, 2, 3) = 3(1, 1, 1) + (-1)(1, 1, 0) + (-1)(1, 0, 0)$$ --- Would you like further details or clarifications? Here are some related questions you might find helpful: 1. How do you express higher-degree polynomials as linear combinations of basis vectors? 2. What does it mean for a vector to be in a span of other vectors? 3. How can you determine if a set of vectors is linearly independent? 4. What is the geometric interpretation of a span of vectors in $$\mathbb{R}^n$$? 5. How would you solve for coefficients in a linear combination using matrices? **Tip**: Always write out the vector components clearly when performing linear combination problems. For polynomials, make sure you match powers of $$x$$, and for vectors, compare corresponding components.