Let x^T A x be a quadratic form in the variables x1, x2, …, xn and define T: R^n →R by T(𝐱)=𝐱^T

step 1 we have that for a t x plus y is equal to x plus y transposed a x plus y that this equals x tran

Let x^T A x be a quadratic form in the variables x1, x2, …,...

Let $x^{T} A x$ be a quadratic form in the variables $x_{1}, x_{2}, \ldots, x_{n}$ and define $T: R^{n} \rightarrow R$ by $T(\mathbf{x})=\mathbf{x}^{T} A \mathbf{x}$. (a) Show that $T(\mathbf{x}+\mathbf{y})=T(\mathbf{x})+2 \mathbf{x}^{T} A \mathbf{y}+T(\mathbf{y})$. (b) Show that $T(c \mathbf{x})=c^{2} T(\mathbf{x})$.

Let $x^{T} A x$ be a quadratic form in the variables $x_{1}, x_{2}, \ldots, x_{n}$ and define $T: R^{n} \rightarrow R$ by $T(\mathbf{x})=\mathbf{x}^{T} A \mathbf{x}$.
(a) Show that $T(\mathbf{x}+\mathbf{y})=T(\mathbf{x})+2 \mathbf{x}^{T} A \mathbf{y}+T(\mathbf{y})$.
(b) Show that $T(c \mathbf{x})=c^{2} T(\mathbf{x})$.

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Key Concepts

Quadratic Forms

A quadratic form is an expression that involves a symmetric arrangement of products of variables, typically represented as x?Ax, where A is a matrix. This concept is essential since it describes a second-degree homogeneous polynomial, and its properties play a pivotal role in understanding geometry, optimization, and multivariable calculus.

Expansion of Quadratic Forms

Expanding a quadratic form of the sum of vectors involves applying the distributive property to matrix multiplication. When evaluating (x + y)?A(x + y), the expression splits into individual quadratic terms and cross terms, revealing that the overall expression is composed of the quadratic forms of each vector plus mixed or interaction terms. This expansion is a fundamental algebraic tool in manipulating and simplifying quadratic expressions.

Homogeneity in Quadratic Forms

The property of homogeneity in a quadratic form indicates that multiplying the input vector by a scalar c results in the quadratic form being multiplied by c², i.e., T(c x) = c²T(x). This characteristic is crucial as it reflects the degree-two nature of the form, ensuring that the scaling behavior is consistent with the underlying polynomial degree.

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