Let's break down the questions visible in the image:

Question 4

This question is asking whether a unique linear combination of two vectors can replace the three-vector combination discussed in a previous question. Specifically, it suggests leaving the combination of the first vector, $a\vec{v}$, unchanged and finding another vector $\vec{w}$, such that when added to $\vec{v}$, it would yield the desired vector (which seems to be given by coordinates $[7.9, 8.7]$).

The problem asks for a comparison between linear combinations involving three vectors and those involving two vectors. To understand the situation:

• Three Vectors: In $\mathbb{R}^2$, it's typical that two vectors are sufficient to express any point in the plane, as long as they are linearly independent. Adding a third vector would be redundant (i.e., the third vector could be written as a combination of the first two).
• Two Vectors: The question explores whether two vectors can still form the desired vector, and asks you to find a specific second vector $\vec{w}$ so that $\vec{v}$ and $\vec{w}$ together still achieve the vector $[7.9, 8.7]$.

Question 5

The question asks how many vectors are needed in $\mathbb{R}^2$ to reach any vector using linear combinations. The key points:

• In $\mathbb{R}^2$, you only need two linearly independent vectors to form any vector in the plane.
• These two vectors must be linearly independent, meaning that one cannot be a scalar multiple of the other.

The condition for linear combinations is that any vector in $\mathbb{R}^2$ can be obtained by appropriately scaling and adding these two vectors. Therefore, the minimum number of vectors needed to reach any vector in $\mathbb{R}^2$ is two, provided they are linearly independent.

Would you like a detailed explanation of how to approach the solution for question 4, or do you have other questions related to linear combinations and vector spaces?

Additional Related Questions:

1. What does it mean for two vectors to be linearly independent?
2. How do you calculate the linear combination of two vectors to get a specific result?
3. Can more than two vectors in $\mathbb{R}^2$ be linearly independent?
4. How do you determine if a set of vectors spans $\mathbb{R}^2$?
5. Can you give an example of linearly dependent and independent vectors in $\mathbb{R}^2$?

Tip:

Remember, in any vector space, the number of vectors needed to span the space is equal to the dimension of the space.