Let T: R³ R² be a linear transformation, where R³ and R2 are the vector spaces over the same field R, defined by

T(x1, x2, x3) = (x1, x2). Then Ker(T) =?

a. {(0,0,0)}

b. {(0,0,x) : x ∈ R}

c. {(0,x,0): x ∈ R}

d. {(x,0,0) : x ∈ 

We are given the linear transformation \( T: \mathbb{R}^3 \to \mathbb{R}^2 \) defined by:

\[
T(x_1, x_2, x_3) = (x_1, x_2)
\]

### To find the kernel of \( T \):

The **kernel** of \( T \), denoted \( \text{Ker}(T) \), consists of all vectors \( (x_1, x_2, x_3) \in \mathbb{R}^3 \) that are mapped to the zero vector in \( \mathbb{R}^2 \), i.e.,

\[
T(x_1, x_2, x_3) = (0, 0)
\]

This means that:

\[
T(x_1, x_2, x_3) = (x_1, x_2) = (0, 0)
\]

Thus, for a vector \( (x_1, x_2, x_3) \) to be in the kernel, it must satisfy:

\[
x_1 = 0 \quad \text{and} \quad x_2 = 0
\]

However, there is no condition on \( x_3 \). Therefore, the kernel consists of all vectors of the form \( (0, 0, x_3) \), where \( x_3 \in \mathbb{R} \).

### Conclusion:
The kernel of \( T \) is:

\[
\text{Ker}(T) = \{(0, 0, x_3) : x_3 \in \mathbb{R}\}
\]

This matches option **b**:

\[
\boxed{b. \{(0, 0, x) : x \in \mathbb{R}\}}
\]

Would you like further details or have any questions?

Here are 5 related questions:

1. What is the definition of the image (range) of a linear transformation?
2. How do we compute the rank of a linear transformation like \( T \)?
3. What is the rank-nullity theorem and how does it apply to this problem?
4. Can the kernel of a linear transformation be the whole domain?
5. How does the matrix representation of \( T \) help in finding its kernel?

**Tip:** The kernel of a transformation tells you about the solutions to \( T(v) = 0 \) and reveals which inputs get "collapsed" to zero.

Let T: R³ → R² be a linear transformation, where R³ and R² are vector spaces over the same field R, defined by T(x1, x2, x3) = (x1, x2). Then Ker(T) =?

a. {(0,0,0)}
b. {(0,0,x) : x ∈ R}
c. {(0,x,0): x ∈ R}
d. {(x,0,0) : x ∈ R}

Explore the solution for finding the kernel of the linear transformation T: R³ → R², defined by T(x1, x2, x3) = (x1, x2). This problem involves concepts of linear algebra, such as kernel and the rank-nullity theorem, suitable for undergraduate-level students.

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