Understanding Eigen Spaces and Eigen Vectors

What is an eigen space in the context of linear algebra?

• A space containing all eigen vectors and the zero vector for a given eigen value.

• A space containing only one eigen vector.

• A space containing only the zero vector.

• A space containing all possible vectors in a matrix.

How can the eigen space corresponding to an eigen value be viewed?

• As the column space of the matrix.

• As the determinant of the matrix.

• As the null space of the matrix Lambda I minus A.

• As the row space of the matrix.

What is the first step in finding the eigen spaces of a 2x2 matrix?

• Finding the inverse of the matrix.

• Finding the eigen values of the matrix.

• Calculating the determinant of the matrix.

• Performing row reduction on the matrix.

In the 2x2 matrix example, what does the eigen space corresponding to an eigen value represent geometrically?

• A plane through the origin.

• A line through the origin.

• A point at the origin.

• A hyperplane in space.

What is the significance of the vector -4,1 in the 2x2 matrix example?

• It is a vector that forms a plane with other vectors.

• It is a vector that does not belong to the eigen space.

• It is the only eigen vector for the matrix.

• It is a basis vector for the eigen space corresponding to the eigen value -4.

What is the first step in finding the eigen spaces of a 3x3 matrix?

• Finding the eigen values by calculating the determinant of Lambda I minus A.

• Performing Gaussian elimination on the matrix.

• Finding the inverse of the matrix.

• Calculating the trace of the matrix.

In the 3x3 matrix example, what does the eigen space corresponding to the eigen value -2 consist of?

• A single vector.

• A hyperplane in space.

• A line through the origin.

• A plane through the origin.

How many vectors are needed to form a basis for the eigen space corresponding to the eigen value -2 in the 3x3 matrix example?

• One vector.

• Two vectors.

• Three vectors.

• Four vectors.

What is the geometric interpretation of the eigen space corresponding to the eigen value 4 in the 3x3 matrix example?

• A plane through the origin.

• A line through the origin.

• A point at the origin.

• A hyperplane in space.

What is the role of Gaussian elimination in finding eigen spaces?

• It is used to find the inverse of the matrix.

• It is used to reduce the matrix to row echelon form to find the null space.

• It is used to calculate the determinant of the matrix.

• It is used to find the trace of the matrix.