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To determine if each set \( U \) of vectors is a basis for \( \mathbb{R}^3 \), we need to check two criteria for each set:
1. The vectors in \( U \) must be linearly independent.
2. The vectors in \( U \) must span \( \mathbb{R}^3 \).

These criteria are both satisfied if the set of vectors can be expressed as a matrix that has a determinant not equal to zero (or, equivalently, if the matrix is of full rank, which is 3 for \( \mathbb{R}^3 \)).

Let's analyze each set \( U \) given in the problem:

### Set (a)
\( U = \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix} \right\} \)

Form the matrix and calculate the determinant:
\[
A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 0 & 4 \end{bmatrix}
\]

### Set (b)
\( U = \left\{ \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \right\} \)

Form the matrix and calculate the determinant:
\[
B = \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}
\]

### Set (c)
\( U = \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \

For each set of vectors U, determine if U is a spanning set for the vector space ℝ^3. Is U a basis for ℝ^3? Use your answers to determine the dimension of ℝ^3.

This problem explores determining if a set of vectors forms a basis for the vector space ℝ^3. Learn how to analyze spanning sets, test linear independence, and calculate determinants for matrices. Suitable for undergraduate students studying Linear Algebra.

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