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Chapter 6: Problem 2

For each of the following inner product spaces \(\mathrm{V}\) (over \(F\) ) and linear transformations \(\mathrm{g}: \mathrm{V} \rightarrow F\), find a vector \(y\) such that \(\mathrm{g}(x)=\langle x, y\rangle\) for all \(x \in \mathrm{V}\). (a) \(\mathrm{V}=\mathrm{R}^{3}, \mathrm{~g}\left(a_{1}, a_{2}, a_{3}\right)=a_{1}-2 a_{2}+4 a_{3}\) (b) \(\mathrm{V}=\mathrm{C}^{2}, \mathrm{~g}\left(z_{1}, z_{2}\right)=z_{1}-2 z_{2}\) (c) \(\mathrm{V}=\mathrm{P}_{2}(R)\) with \(\langle f(x), h(x)\rangle=\int_{0}^{1} f(t) h(t) d t, \mathrm{~g}(f)=f(0)+f^{\prime}(1)\)

Short Answer

Expert verified
(a) We find the vector y = (1, -2, 4) for the linear transformation g(a鈧, a鈧, a鈧) = a鈧 - 2a鈧 + 4a鈧 in the inner product space R鲁. (b) We find the vector y = (1, 2) for the linear transformation g(z鈧, z鈧) = z鈧 - 2z鈧 in the inner product space C虏. (c) We find the polynomial y(t) = 2t + 1 for the linear transformation g(f) = f(0) + f'(1) in the inner product space P鈧(R) with 鉄╢(x), h(x)鉄 = 鈭個鹿 f(t)h(t)dt.

Step by step solution

01

Identify the vector space and inner product

The given vector space V is R鲁 and the linear transformation g(a鈧, a鈧, a鈧) = a鈧 - 2a鈧 + 4a鈧.
02

Find the vector y

We need to find y such that g(x) = 鉄▁, y鉄 for all x in V. For two vectors x = (x鈧, x鈧, x鈧) and y = (y鈧, y鈧, y鈧) in R鲁, the inner product is defined as: 鉄▁, y鉄 = x鈧亂鈧 + x鈧倅鈧 + x鈧儁鈧. We want g(x) = 鉄▁, y鉄, which means: x鈧亂鈧 + x鈧倅鈧 + x鈧儁鈧 = x鈧 - 2x鈧 + 4x鈧. Comparing the coefficients of x鈧, x鈧, and x鈧, we can see that y鈧 = 1, y鈧 = -2, and y鈧 = 4. This gives us the vector y = (1, -2, 4). ## Case (b) ##
03

Identify the vector space and inner product

The given vector space V is C虏 and the linear transformation g(z鈧, z鈧) = z鈧 - 2z鈧.
04

Find the vector y

We need to find y such that g(x) = 鉄▁, y鉄 for all x in V. For two vectors x = (x鈧, x鈧) and y = (y鈧, y鈧) in C虏, the inner product is defined as: 鉄▁, y鉄 = x鈧 * conjugate(y鈧) + x鈧 * conjugate(y鈧). We want g(x) = 鉄▁, y鉄, which means: x鈧 * conjugate(y鈧) + x鈧 * conjugate(y鈧) = x鈧 - 2x鈧. Comparing the coefficients of x鈧 and x鈧, we can see that conjugate(y鈧) = 1 and conjugate(y鈧) = -2. This gives us the conjugates of the components of vector y: y* = (1, -2). Taking the complex conjugate of y* to find y, we have y = (1, 2). ## Case (c) ##
05

Identify the vector space and inner product

The given vector space V is P鈧(R) (set of real polynomials of degree at most 2) and the inner product is defined as 鉄╢(x), h(x)鉄 = 鈭個鹿 f(t)h(t)dt. The linear transformation g(f) = f(0) + f'(1).
06

Find the vector y

We need to find a polynomial y(t) such that g(f) = 鉄╢, y鉄 for all f in V. Let f be an arbitrary polynomial in P鈧(R), written as f(t) = at虏 + bt + c. Then, f'(t) = 2at + b. The linear transformation g(f) is given by g(f) = f(0) + f'(1) = c + (2a + b). Now, let y(t) = 伪t虏 + 尾t + 纬 be the desired polynomial in P鈧(R). We want: g(f) = 鉄╢, y鉄 鉄 c + (2a + b) = 鈭個鹿(at虏 + bt + c)(伪t虏 + 尾t + 纬)dt. Expanding and integrating term by term, and comparing the coefficients of the polynomial, we can find the values of 伪, 尾, and 纬: 伪 = 0, 尾 = 2, 纬 = 1. So, the polynomial y(t) = 2t + 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inner Product Spaces
An inner product space is a vector space with an additional structure called the inner product. This inner product is a function that associates each pair of vectors in the space with a scalar, typically denoted as \( \langle x, y \rangle \). It generalizes the dot product from Euclidean space. The inner product allows us to determine angles and lengths and is crucial for defining orthogonality and projections.

Here are some key properties of inner products:
  • **Conjugate Symmetry**: \( \langle x, y \rangle = \overline{\langle y, x \rangle} \). This means that swapping the vectors changes the sign of the inner product.
  • **Linearity in the First Argument**: \( \langle ax + by, z \rangle = a\langle x, z \rangle + b\langle y, z \rangle \) for scalars \( a \) and \( b \).
  • **Positive Definiteness**: \( \langle x, x \rangle \geq 0 \) and \( \langle x, x \rangle = 0 \) if and only if \( x = 0 \).
Inner product spaces are foundational to many areas of mathematics and physics, providing tools to study geometry and orthogonality within abstract vector spaces.
Vector Spaces
A vector space is a collection of objects called vectors, which can be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often real or complex numbers. Vector spaces are defined by a set of axioms, including the ability to add vectors and multiply vectors by scalars, resulting in another vector within the same space.

Important aspects of vector spaces include:
  • **Closure under Addition**: If \( x \) and \( y \) are vectors in the space, then \( x + y \) is also in the space.
  • **Closure under Scalar Multiplication**: For a vector \( x \) and scalar \( a \), the product \( ax \) is also in the space.
  • **Existence of Additive Identity**: There exists a zero vector \( 0 \) such that \( x + 0 = x \) for any vector \( x \).
  • **Existence of Additive Inverses**: For every vector \( x \), there exists a vector \( y \) such that \( x + y = 0 \).
Understanding these fundamental properties helps in analyzing mathematical models across various fields, including physics and computer science.
Polynomial Functions
Polynomial functions are expressions composed of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. They are a central part of algebra and have applications in calculus and differential equations.

A polynomial function of degree \( n \) can be expressed in the form:
  • \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)
where \( a_n, a_{n-1}, \ldots, a_0 \) are coefficients and \( x \) is the variable.

Some important features of polynomial functions include:
  • **Degree of a Polynomial**: The degree is the highest power of the variable in the polynomial. It determines the polynomial's general shape and behavior.
  • **Roots and Zeroes**: The solutions to \( f(x) = 0 \) are called roots or zeroes, which are the points where the graph of the polynomial touches or crosses the x-axis.
  • **Synthetic Division**: A simplified method of dividing polynomials, useful especially for finding roots and simplifying calculations.
Polynomials have straightforward graphs, and because of their smooth, continuous nature, they play a critical role in approximation theory and numerical analysis.

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Most popular questions from this chapter

Let \(\mathrm{V}\) be a real inner product space. (a) Prove that any translation on \(V\) is a rigid motion. (b) Prove that the composite of any two rigid motions on \(V\) is a rigid motion on V.

In \(\mathrm{R}^{2}\), let $$ \beta=\left\\{\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right),\left(\frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}}\right)\right\\} . $$ Find the Fourier coefficients of \((3,4)\) relative to \(\beta\).

Let \(V\) be a complex inner product space, and let \(T\) be a linear operator on \(\mathrm{V}\). Define $$ \mathrm{T}_{1}=\frac{1}{2}\left(\mathrm{~T}+\mathrm{T}^{*}\right) \quad \text { and } \quad \mathrm{T}_{2}=\frac{1}{2 i}\left(\mathrm{~T}-\mathrm{T}^{*}\right) . $$ (a) Prove that \(\mathrm{T}_{1}\) and \(\mathrm{T}_{2}\) are self-adjoint and that \(\mathrm{T}=\mathrm{T}_{1}+i \mathrm{~T}_{2}\). (b) Suppose also that \(\mathrm{T}=\mathrm{U}_{1}+i \mathrm{U}_{2}\), where \(\mathrm{U}_{1}\) and \(\mathrm{U}_{2}\) are self-adjoint. Prove that \(\mathrm{U}_{1}=\mathrm{T}_{1}\) and \(\mathrm{U}_{2}=\mathrm{T}_{2}\). (c) Prove that \(T\) is normal if and only if \(T_{1} T_{2}=T_{2} T_{1}\).

For each of the given quadratic forms \(K\) on a real inner product space \(\mathrm{V}\), find a symmetric bilinear form \(H\) such that \(K(x)=H(x, x)\) for all \(x \in \mathrm{V}\). Then find an orthonormal basis \(\beta\) for \(\mathrm{V}\) such that \(\psi_{\beta}(H)\) is a diagonal matrix. (a) \(K: \mathrm{R}^{2} \rightarrow R\) defined by \(K\left(\begin{array}{l}t_{1} \\\ t_{2}\end{array}\right)=-2 t_{1}^{2}+4 t_{1} t_{2}+t_{2}^{2}\) (b) \(K: \mathrm{R}^{2} \rightarrow R\) defined by \(K\left(\begin{array}{l}t_{1} \\\ t_{2}\end{array}\right)=7 t_{1}^{2}-8 t_{1} t_{2}+t_{2}^{2}\) (c) \(K: \mathrm{R}^{3} \rightarrow R\) defined by \(K\left(\begin{array}{l}t_{1} \\\ t_{2} \\ t_{3}\end{array}\right)=3 t_{1}^{2}+3 t_{2}^{2}+3 t_{3}^{2}-2 t_{1} t_{3}\)

Let \(\mathrm{T}\) be a normal operator on a finite-dimensional complex inner product space \(\mathrm{V}\). Use the spectral decomposition \(\lambda_{1} \mathrm{~T}_{1}+\lambda_{2} \mathrm{~T}_{2}+\cdots+\lambda_{k} \mathrm{~T}_{k}\) of \(\mathrm{T}\) to prove the following results. (a) If \(g\) is a polynomial, then $$ g(\mathbf{T})=\sum_{i=1}^{k} g\left(\lambda_{i}\right) \mathbf{T}_{i} . $$ (b) If \(\mathrm{T}^{n}=\mathrm{T}_{0}\) for some \(n\), then \(\mathrm{T}=\mathrm{T}_{0}\). (c) Let \(U\) be a linear operator on \(V\). Then \(U\) commutes with \(T\) if and only if \(U\) commutes with each \(T_{i}\). (d) There exists a normal operator \(U\) on \(V\) such that \(U^{2}=T\). (e) \(\mathrm{T}\) is invertible if and only if \(\lambda_{i} \neq 0\) for \(1 \leq i \leq k\). (f) \(\mathrm{T}\) is a projection if and only if every eigenvalue of \(\mathrm{T}\) is 1 or 0 . (g) \(\mathrm{T}=-\mathrm{T}^{*}\) if and only if every \(\lambda_{i}\) is an imaginary number.
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