To describe the geometric set of all linear combinations of the given vectors in $\mathbb{R}^3$, we need to determine the span of each set of vectors. The span of a set of vectors can describe a point, a line, a plane, or all of $\mathbb{R}^3$, depending on the linear independence of the vectors.

Let's analyze each set:

(a) $\{(1, -1, 1), (1, 2, 0), (4, 5, 1)\}$

To determine the geometric description, we first check if these vectors are linearly independent by finding the determinant of the matrix formed by these vectors as columns.

$\text{Matrix } A = \begin{pmatrix} 1 & 1 & 4 \\ -1 & 2 & 5 \\ 1 & 0 & 1 \end{pmatrix}$

Calculate the determinant:

$\text{det}(A) = 1 \cdot (2 \cdot 1 - 5 \cdot 0) - 1 \cdot (-1 \cdot 1 - 5 \cdot 1) + 4 \cdot (-1 \cdot 0 - 2 \cdot 1)$

$= 1 \cdot 2 - 1 \cdot (-6) + 4 \cdot (-2)$

$= 2 + 6 - 8 = 0$

Since the determinant is zero, the vectors are linearly dependent. Therefore, the set of all linear combinations of these vectors forms a plane in $\mathbb{R}^3$.

(b) $\{(1, 2, -3), (-2, -4, 6), (-3, -6, 9)\}$

Examine the linear dependence of these vectors. Notice that:

$(-2, -4, 6) = -2 \cdot (1, 2, -3)$ $(-3, -6, 9) = -3 \cdot (1, 2, -3)$

All vectors are scalar multiples of each other, meaning they are linearly dependent. The span of these vectors is a line in $\mathbb{R}^3$, as they are all collinear.

(c) $\{(2, -1, 0), (-1, 2, -1), (0, -1, 2)\}$

Again, let's check for linear independence by calculating the determinant of the matrix formed by these vectors:

$\text{Matrix } B = \begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{pmatrix}$

Calculate the determinant:

$\text{det}(B) = 2 \cdot (2 \cdot 2 - (-1) \cdot (-1)) - (-1) \cdot (-1 \cdot 2 - (-1) \cdot 0) + 0 \cdot ((-1) \cdot (-1) - 2 \cdot 0)$

$= 2 \cdot (4 - 1) - (-1) \cdot (-2)$

$= 2 \cdot 3 - 2 = 6 - 2 = 4$

Since the determinant is non-zero, these vectors are linearly independent. Therefore, the span of these vectors is all of $\mathbb{R}^3$.

Summary

• (a) The set of all linear combinations forms a plane in $\mathbb{R}^3$.
• (b) The set of all linear combinations forms a line in $\mathbb{R}^3$.
• (c) The set of all linear combinations spans all of $\mathbb{R}^3$.

Would you like more details on any part of this explanation, or do you have any questions?

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Here are 5 related questions to expand on this topic:

1. How do you determine if vectors in $\mathbb{R}^n$ are linearly dependent or independent?
2. What does the span of a set of vectors represent geometrically?
3. Can you explain the significance of the determinant in relation to vector independence?
4. What are the possible geometric interpretations of the span of vectors in $\mathbb{R}^2$?
5. How does the concept of linear independence extend to higher dimensions beyond $\mathbb{R}^3$?

Tip: When calculating determinants for larger matrices, breaking down the determinant into smaller parts using cofactor expansion can simplify the process.