Project Using the Transformation

Problem

A=[[1,0,1],[1,−1,1],[0,1,1]]

Solution

1. Identify the task as finding the projection matrix P that projects vectors onto the subspace spanned by the given basis vectors (a_1) (a_2) and (a_3)

2. Construct the matrix A using the given vectors as columns.

A=[[1,0,1],[1,−1,1],[0,1,1]]

3. Calculate the determinant of A to determine if the vectors are linearly independent and span ℝ3

det(A)=1*(−1−1)−0*(1−0)+1*(1−0)

det(A)=−2+1=−1

4. Conclude that since the determinant is non-zero, the vectors (a_1),(a_2),(a_3) form a basis for ℝ3

5. Apply the definition of a projection onto the entire space. A projection onto the space spanned by a basis of ℝ3 is simply the identity transformation.

P=A*(AT*A)(−1)*AT=I

Final Answer

P=[[1,0,0],[0,1,0],[0,0,1]]

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