Math Problem Statement
What is the matrix for projW : R
3 → R
3
, where
W = Span
1
0
−1
,
1
1
1
Solution
To find the projection matrix for onto the subspace spanned by the vectors
we first construct a matrix whose columns are these vectors:
The projection matrix onto the column space of is given by:
Now, we need to compute :
Next, we find :
Now we can calculate :
Calculating :
Now, we can compute the full projection matrix :
After some calculations, the final projection matrix simplifies to:
Summary
The projection matrix for is:
Would you like further details or have any questions about this process? Here are some related questions you might consider:
- How do you derive the orthogonal projection formula in general?
- What are the geometric interpretations of projection matrices?
- How would you compute the projection of a specific vector onto the subspace ?
- Can you explain the significance of the rank of the matrix in this context?
- What would happen if we added a third vector to the spanning set of ?
Tip: Always check that the projection matrix is idempotent (i.e., ).
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Projection Matrices
Vector Spaces
Formulas
P = A(A^TA)^{-1}A^T
Theorems
Properties of Projection Matrices
Suitable Grade Level
Grades 11-12
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