
*2 points*

Let the set {(a,b),(c,d)}{(*a*,*b*),(*c*,*d*)} be a linearly independent subset of R2R2. Choose the set of correct options.

{(a,b,0),(c,d,0)}{(*a*,*b*,0),(*c*,*d*,0)} must be a linearly independent subset of R3R3.

{(a,b,0),(c,d,0),(0,0,1)}{(*a*,*b*,0),(*c*,*d*,0),(0,0,1)} must be a linearly independent subset of R3R3.

{(a,b,0),(c,d,0),(1,0,0)}{(*a*,*b*,0),(*c*,*d*,0),(1,0,0)} must be a linearly independent subset of R3R3.

{(a,b,0),(c,d,0),(0,1,0)}{(*a*,*b*,0),(*c*,*d*,0),(0,1,0)} must be a linearly independent subset of R3R3


Let’s analyze each option in detail, keeping in mind that a set of vectors is linearly independent if the only solution to the linear combination equaling the zero vector is when all the coefficients are zero.

Given that \(\{(a,b), (c,d)\}\) is linearly independent in \(\mathbb{R}^2\), we know that there are no non-trivial solutions to the equation:

\[
x_1(a, b) + x_2(c, d) = (0, 0)
\]
which implies that \(x_1 = x_2 = 0\).

Now, we consider each set in \(\mathbb{R}^3\):

### 1. \(\{(a,b,0), (c,d,0)\}\)
In \(\mathbb{R}^3\), this is equivalent to the following equation for linear dependence:

\[
x_1(a, b, 0) + x_2(c, d, 0) = (0, 0, 0)
\]
The third component for both vectors is 0, meaning the problem reduces to checking the linear independence of \((a,b)\) and \((c,d)\) in \(\mathbb{R}^2\), which we know is already linearly independent. Thus, \(\{(a,b,0), (c,d,0)\}\) is linearly independent in \(\mathbb{R}^3\).

**Conclusion**: This set **must be linearly independent**.

### 2. \(\{(a,b,0), (c,d,0), (0,0,1)\}\)
Now we check if adding the vector \((0,0,1)\) preserves linear independence:

\[
x_1(a, b, 0) + x_2(c, d, 0) + x_3(0, 0, 1) = (0, 0, 0)
\]
The third component forces \(x_3 = 0\). This reduces the problem to:

\[
x_1(a, b) + x_2(c, d) = (0, 0)
\]
Since \((a,b)\) and \((c,d)\) are linearly independent, we conclude that \(x_1 = x_2 = 0\). Therefore, the entire set is linearly independent.

**Conclusion**: This set **must be linearly independent**.

### 3. \(\{(a,b,0), (c,d,0), (1,0,0)\}\)
Next, consider the vector \((1,0,0)\):

\[
x_1(a, b, 0) + x_2(c, d, 0) + x_3(1, 0, 0) = (0, 0, 0)
\]
The third component forces \(x_1 = x_2 = 0\). Then we have \(x_3(1, 0, 0) = (0, 0, 0)\), which implies \(x_3 = 0\).

Thus, all coefficients are zero, so the set is linearly independent.

**Conclusion**: This set **must be linearly independent**.

### 4. \(\{(a,b,0), (c,d,0), (0,1,0)\}\)
Finally, consider the vector \((0,1,0)\):

\[
x_1(a, b, 0) + x_2(c, d, 0) + x_3(0, 1, 0) = (0, 0, 0)
\]
The third component forces \(x_1 = x_2 = 0\), and the second component forces \(x_3 = 0\). Hence, the set is linearly independent.

**Conclusion**: This set **must be linearly independent**.

### Final Answer:
The following sets **must be linearly independent** in \(\mathbb{R}^3\):
- \(\{(a,b,0), (c,d,0)\}\)
- \(\{(a,b,0), (c,d,0), (0,0,1)\}\)
- \(\{(a,b,0), (c,d,0), (1,0,0)\}\)
- \(\{(a,b,0), (c,d,0), (0,1,0)\}\)

Do you need more details or have any questions? Here are some follow-up questions you might consider:

1. How do we define linear independence in higher-dimensional spaces?
2. Why is the vector \((0, 0, 1)\) in \(\mathbb{R}^3\) automatically independent of the other two vectors?
3. Can a set of more than two vectors in \(\mathbb{R}^2\) ever be linearly independent?
4. How can we generalize the idea of linear independence to larger sets of vectors in \(\mathbb{R}^n\)?
5. What happens to linear independence if one of the vectors is a linear combination of the others?

**Tip**: Always check the third components in \(\mathbb{R}^3\) when vectors are extended from \(\mathbb{R}^2\). If they’re non-zero and the original vectors are independent, you are likely maintaining independence.

Let the set {(a,b),(c,d)} be a linearly independent subset of R2. Choose the set of correct options: {(a,b,0),(c,d,0)} must be a linearly independent subset of R3. {(a,b,0),(c,d,0),(0,0,1)} must be a linearly independent subset of R3. {(a,b,0),(c,d,0),(1,0,0)} must be a linearly independent subset of R3. {(a,b,0),(c,d,0),(0,1,0)} must be a linearly independent subset of R3.

Explore linear independence in R2 and R3 with a step-by-step analysis of different vector sets, including {(a,b,0),(c,d,0)} and {(a,b,0),(c,d,0),(0,0,1)}. Learn how to determine whether these sets are linearly independent in R3, based on fundamental theorems and vector space properties. Suitable for college-level linear algebra students.

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