Use part (a) to compute the vector [tex]\text{proj}_{\text{col}(\mathbf{a})}\mathbf{b}[/tex], i.e., the orthogonal projection of [tex]\mathbf{b}[/tex] onto the column space of [tex]\mathbf{a}[/tex]. Enter the vector [tex]\text{proj}_{\text{col}(\mathbf{a})}\mathbf{b}[/tex] in the form [tex][c_1, c_2, c_3][/tex]:

Question

13-02-2025
Mathematics

High School

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Use part (a) to compute the vector [tex]\text{proj}_{\text{col}(\mathbf{a})}\mathbf{b}[/tex], i.e., the orthogonal projection of [tex]\mathbf{b}[/tex] onto the column space of [tex]\mathbf{a}[/tex]. Enter the vector [tex]\text{proj}_{\text{col}(\mathbf{a})}\mathbf{b}[/tex] in the form [tex][c_1, c_2, c_3][/tex]:

Answer :

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pankajprajapati9091 pankajprajapati9091 · Answer 1 · 2023-08-17T02:07:20-04:00

Final answer:

The question involves computing the orthogonal projection of vector b onto the column space of matrix or vector a using the projection operator formula where PAB is the projection of B onto A, and involves dot products and scaling.

Explanation:

The student's question is related to finding the orthogonal projection of a vector b onto the column space of another vector or matrix a. In linear algebra, this is accomplished using the formula for the projection operator given a column vector A and a column vector B, the orthogonal projection of B onto A is given by the expression PAB = (A⋅B)/(A⋅A) × A, where ⋅ denotes the dot product. Considering the column space of a matrix a essentially involves computing the projection onto each column vector of a and summing the results, assuming the columns of a are orthogonal.

If we have specific vectors A and B, and A represents the column space of matrix a, we can apply this formula to get the coordinates [c1, c2, c3] for the projection of B onto A. The detailed steps would require calculating the dot products and the scaling according to the given formula.