Find the Null Space

Problem

Null Space*({[[2],[−1],[3],[4],[2]],[[6],[2],[−1],[3],[4]]}+[[0],[1],[4],[4],[3]])

Solution

1. Identify the nature of the set. The expression describes an affine subspace, which is a translation of a linear subspace spanned by two vectors. However, the "Null Space" operation is typically defined for a matrix or a linear transformation, not an affine set.

2. Interpret the task as finding the null space of the matrix A whose columns are the given basis vectors. In linear algebra, the null space of a set of vectors is the null space of the matrix formed by those vectors as rows or columns. Given the context of finding a "Null Space" for a set of vectors, we treat the vectors as the columns of a matrix A

3. Construct the matrix A using the two basis vectors as columns. The translation vector [0,1,4,4,3]T does not affect the null space of the linear transformation itself.

A=[[2,6],[−1,2],[3,−1],[4,3],[2,4]]

4. Set up the equation A*x=0 to find the null space. Here, x must be a vector in ℝ2 because A has 2 columns.

[[2,6],[−1,2],[3,−1],[4,3],[2,4]]*[[(x_1)],[(x_2)]]=[[0],[0],[0],[0],[0]]

5. Analyze the columns of A We check if the columns are linearly independent. If the only solution to (x_1)*(v_1)+(x_2)*(v_2)=0 is (x_1)=(x_2)=0 the null space contains only the zero vector.

6. Test for linear independence. Looking at the first two rows:

2*(x_1)+6*(x_2)=0

−(x_1)+2*(x_2)=0⇒(x_1)=2*(x_2)

7. Substitute (x_1)=2*(x_2) into the first row equation:

2*(2*(x_2))+6*(x_2)=0

4*(x_2)+6*(x_2)=0

10*(x_2)=0⇒(x_2)=0

8. Conclude that since (x_2)=0 then (x_1)=2*(0)=0 The columns are linearly independent.

Final Answer

Null Space*([2,6],[−1,2],[3,−1],[4,3],[2,4])={[0],[0]}

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