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To find the basis for the eigenspace corresponding to the eigenvalue \(\lambda = 1\) for the matrix \(A\), we need to solve the equation \((A - \lambda I)\mathbf{x} = 0\).

Given:
\[ A = \begin{pmatrix} 1 & 0 \\ -3 & 4 \end{pmatrix}, \quad \lambda = 1 \]

First, calculate \(A - \lambda I\):
\[ A - \lambda I = \begin{pmatrix} 1 & 0 \\ -3 & 4 \end{pmatrix} - \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ -3 & 3 \end{pmatrix} \]

Now, we need to find the null space of the matrix \(\begin{pmatrix} 0 & 0 \\ -3 & 3 \end{pmatrix}\).

Set up the equation:
\[\begin{pmatrix} 0 & 0 \\ -3 & 3 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}\]

This simplifies to:
\[ -3x_1 + 3x_2 = 0 \]

From this equation, we get:
\[ x_1 = x_2 \]

Therefore, the basis for the eigenspace corresponding to \(\lambda = 1\) is given by the vector:
\[ \begin{pmatrix} 1 \\ 1 \end{pmatrix} \]

So, the answer is:
\[ \left\{ \begin{pmatrix} 1 \\ 1 \end{pmatrix} \right\} \]

Would you like a detailed explanation or have any questions?

Here are some related questions:
1. What is an eigenvalue?
2. How is the eigenspace related to the eigenvalue?
3. What are the steps to find eigenvectors?
4. How do you verify if a vector is an eigenvector of a matrix?
5. What is the significance of the basis in an eigenspace?
6. Can there be more than one basis vector for an eigenspace?
7. What does it mean if a matrix has repeated eigenvalues?
8. How does one determine the multiplicity of an eigenvalue?

**Tip:** Always check if the calculated eigenvectors are linearly independent to confirm the basis set for the eigenspace.

Discover how to find the basis for the eigenspace corresponding to eigenvalue λ = 1 for matrix A = [1 0; -3 4]. Step-by-step solution including the calculation of A - λI and solving the null space equation. Explore the significance of eigenspaces and basis vectors in linear algebra.

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