Math Problem Statement
I don't understand how you expanded the quadratic expression?
Solution
Certainly! Let's carefully go through the expansion of the quadratic expression.
Given Expression:
We start with the expression given in the problem: Here, is a scalar, and and are row vectors in . The notation represents the transpose of the row vector , making it a column vector.
Expansion of the Expression:
Let's expand step by step.
-
First Term: Here, is a scalar (since it's a row vector multiplied by its transpose).
-
Second Term: This is a scalar since it is the dot product of two vectors.
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Third Term: Again, this is a scalar.
-
Fourth Term: This term is also a scalar.
Full Expansion:
Now, let's put it all together:
But note that and are actually the same scalar, so we can combine these terms:
This is a quadratic expression in , with:
- (coefficient of ),
- (coefficient of ),
- (constant term).
Summary:
Thus, the expression expands into the quadratic form: This is the basis for using the discriminant to establish the inequality.
Would you like to explore further or clarify any other steps?
Related Questions:
- Why does the quadratic expression need to be non-negative?
- How do the vectors and interact geometrically in this inequality?
- What role does the discriminant play in determining the nature of quadratic equations?
- Can this proof technique be applied to other forms of inequalities?
- What is the significance of the Cauchy-Schwarz inequality in inner product spaces?
Tip:
When expanding expressions like this, carefully track each term and how they combine. Visualizing vector multiplication can help you understand the nature of each term.
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Math Problem Analysis
Mathematical Concepts
Quadratic Expressions
Vector Algebra
Matrix Algebra
Formulas
Quadratic expression expansion
Theorems
Cauchy-Schwarz inequality
Suitable Grade Level
Advanced High School
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