Math Problem Statement
Let a,b∈R and let u , v , w∈Rn , where v≠0 . Show that
projv(au+bw)=a(projvu)+b(projvw)
Solution
To show that
where , , and , we will use the definition of the projection of a vector onto another vector.
Step 1: Recall the definition of the projection
The projection of a vector onto a non-zero vector is given by:
where is the dot product between and , and is the dot product of with itself.
Step 2: Apply the projection to the left-hand side
We want to compute . Using the definition of the projection:
Step 3: Expand the dot product
The dot product is linear, so we can distribute it over the sum:
Thus, we have:
Step 4: Separate the terms
We can now separate the terms involving and :
Step 5: Recognize the projections
From the definition of the projection, we recognize the terms on the right-hand side as projections:
Thus, we have shown that:
Conclusion
We have successfully demonstrated that the projection operation is linear with respect to scalar multiplication and vector addition.
Would you like any further clarification on any steps or more details?
Here are 5 related questions:
- How does the projection formula change if is a unit vector?
- What properties must a vector satisfy for the projection operation to be well-defined?
- Can we extend this result to complex vectors? How does the dot product change?
- What is the geometric interpretation of the projection formula?
- How does this projection relate to the concept of orthogonal decomposition?
Tip: The projection of a vector onto another vector minimizes the distance between the original vector and the projected vector along the direction of the second vector.
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Math Problem Analysis
Mathematical Concepts
Vector Projections
Linear Algebra
Dot Product
Vector Spaces
Formulas
Projection of a vector u onto v: projv(u) = (u ⋅ v / v ⋅ v) v
Linear property of dot product: (a*u + b*w) ⋅ v = a*(u ⋅ v) + b*(w ⋅ v)
Theorems
Linearity of Projections
Properties of Dot Product
Suitable Grade Level
Undergraduate (Linear Algebra)