Question

a. Show that \(\mathbb{R}^{3}\) is spanned by \\{(1,0,1),(1,1,0),(0,1,1)\\} b. Show that \(\mathbf{P}_{2}\) is spanned by \(\left\\{1+2 x^{2}, 3 x, 1+x\right\\}\). c. Show that \(\mathbf{M}_{22}\) is spanned by \(\left\\{\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right],\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\right\\}\)

Short answer

Every vector, polynomial, and matrix in the respective spaces can be expressed as a linear combination of the given sets; thus, they span the spaces.

Step-by-step solution

Define Vectors in \(\mathbb{R}^3\)

We are given the vectors \(\mathbf{v}_1 = (1, 0, 1)\), \(\mathbf{v}_2 = (1, 1, 0)\), and \(\mathbf{v}_3 = (0, 1, 1)\). We need to show that every vector \(\mathbf{v} = (x, y, z)\) in \(\mathbb{R}^3\) can be written as a linear combination of these vectors.

Set Up the Linear Combination

Express a general vector \(\mathbf{v} = (x, y, z)\) as a linear combination of \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\). This gives the equation \(a(1, 0, 1) + b(1, 1, 0) + c(0, 1, 1) = (x, y, z)\).

Solve System of Equations for \(\mathbb{R}^3\)

Write the equation from Step 2 component-wise: 1. \(a + b = x\)2. \(b + c = y\)3. \(a + c = z\)Solve these equations simultaneously to find \(a, b,\) and \(c\) for any \(x, y, z\).

Conclude Part 'a'

The solution to the linear equations shows that \(\mathbb{R}^3\) is spanned by \{(1,0,1),(1,1,0),(0,1,1)\}.

Define Polynomials in \(\mathbf{P}_2\)

We are given the polynomials \(p_1(x) = 1 + 2x^2\), \(p_2(x) = 3x\), and \(p_3(x) = 1 + x\). We need to show that any polynomial \(p(x) = ax^2 + bx + c\) in \(\mathbf{P}_2\) can be expressed as a linear combination of these polynomials.

Set Up the Linear Combination for Polynomials

Express a general polynomial \(p(x) = ax^2 + bx + c\) as a linear combination \(d(1 + 2x^2) + e(3x) + f(1 + x)\).

Solve for \(d, e, f\)

Equate the coefficients from the linear combination setup:1. \(2d = a\)2. \(3e + f = b\)3. \(d + f = c\)Solve this system of equations.

Conclude Part 'b'

Since the coefficients match up for arbitrary constants, \(\mathbf{P}_{2}\) is spanned by \{1+2x^{2}, 3x, 1+x\}.

Define Matrices in \(\mathbf{M}_{22}\)

We are given the matrices \(A_1 = \left[\begin{array}{cc}1 & 0 \ 0 & 0\end{array}\right]\), \(A_2 = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right]\), \(A_3 = \left[\begin{array}{cc}0 & 1 \ 1 & 0\end{array}\right]\), \(A_4 = \left[\begin{array}{cc}1 & 1 \ 0 & 1\end{array}\right]\). We need to show that any matrix \(\left[\begin{array}{cc} a & b \ c & d \end{array}\right]\) in \(\mathbf{M}_{22}\) can be expressed as a linear combination of these matrices.

Set Up the Linear Combination for Matrices

Express a general matrix \(\left[\begin{array}{cc} a & b \ c & d \end{array}\right]\) using \(k_1A_1 + k_2A_2 + k_3A_3 + k_4A_4\) to obtain:\(\left[\begin{array}{cc} k_1 + k_2 + k_4 & k_3 + k_4 \ k_3 & k_2 + k_4 \end{array}\right] = \left[\begin{array}{cc} a & b \ c & d \end{array}\right]\)

Solve for \(k_1, k_2, k_3, k_4\)

Write the matrix equation component-wise and solve for the variables:1. \(k_1 + k_2 + k_4 = a\)2. \(k_3 + k_4 = b\)3. \(k_3 = c\)4. \(k_2 + k_4 = d\).

Conclude Part 'c'

With solutions existing for arbitrary \(a, b, c, d\), the space \(\mathbf{M}_{22}\) is spanned by the given matrices.

Key concepts

Linear Algebra

Linear algebra is a branch of mathematics that is primarily concerned with vectors, vector spaces, and the transformations between these spaces. This field is fundamental in both applied and theoretical mathematics and is essential in various scientific domains including physics, engineering, and computer science. At its core, linear algebra studies concepts like vectors which are entities that have both magnitude and direction. Alongside vectors, matrices are grid-like structures of numbers or functions that can represent a set of equations or transformations.

• Vectors: Objects representing quantities that possess both magnitude and direction, useful for representing real-world phenomena like forces or velocities.
• Matrices: Rectangular arrays of numbers or expressions that facilitate operations such as solving systems of equations and transforming geometric data.

Understanding the principles of linear algebra is crucial when dealing with multi-dimensional spaces, optimizing systems, and in the analysis of algorithms.

Basis and Span

In linear algebra, the concepts of basis and span are pivotal when discussing vector spaces. A vector space is essentially a collection of vectors that can be added together or multiplied by scalars to yield another vector in the same space. The span refers to the set of all possible vectors that can be expressed as a combination of given vectors. For instance, in ext{R}^3, a set of vectors might span the entire space if every vector in ext{R}^3 can be represented as their combination.

• Span: Refers to the collection of all vectors that can be constructed from linear combinations of a given set.
• Basis: A set of vectors that is both spanning and linearly independent, meaning no vector in the basis can be written as a combination of the others.

In essence, if a set of vectors spans a space and is chosen appropriately, it serves as a basis, providing a foundation to express any vector in that space.

Linear Combination

A linear combination involves forming new vectors by multiplying given vectors by scalars and adding the results. This concept forms the backbone of many operations in linear algebra. When you write a vector as a linear combination of other vectors, you effectively express it using their weighted sum. This expression is crucial for determining if a set of vectors can span a vector space.

• Linear Combination: Given vectors ext{v}_1, ext{v}_2, ..., ext{v}_n, a linear combination is any expression of the form a_1 ext{v}_1 + a_2 ext{v}_2 + ... + a_n ext{v}_n, where a_1, a_2, ..., a_n are scalars.
• Expressing vectors: If a vector can be expressed as a linear combination of a set of vectors, those vectors span the space.

Linear combinations are integral in solving systems of linear equations and are evident in techniques such as Gaussian elimination.

Matrix Representation

Matrices play a key role in representing linear transformations and systems of equations, serving as compact and efficient means to handle large datasets and complex geometric transformations. A matrix can represent the coefficients of vectors in a system of linear equations or denote a transformation applied to a space.

• Matrix: An array of numbers organized in rows and columns that facilitates transformations and computations.
• Linear Transformation: A function from one vector space to another utilizing matrices, preserving operations like addition and scalar multiplication.

These structures simplify calculations and enable the exploration of vector spaces' properties. Mastery of matrix operations allows for solving linear systems, finding invertible matrices, and understanding dimensional transformations.