Question

Show that each of the following sets of vectors is independent. a. \(\left\\{1+x, 1-x, x+x^{2}\right\\}\) in \(\mathbf{P}_{2}\) b. \(\left\\{x^{2}, x+1,1-x-x^{2}\right\\}\) in \(\mathbf{P}_{2}\) \(\left\\{\left[\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 1 & 0\end{array}\right],\left[\begin{array}{rr}0 & 0 \\ 1 & -1\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 0 & 1\end{array}\right]\right\\}\) \(\left\\{\left[\begin{array}{ll}1 & 1 \\ 1 & 0 \\ \text { in } & \mathbf{M}_{22}\end{array}\right],\left[\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right],\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]\right\\}\)

Short answer

All sets of vectors are independent.

Step-by-step solution

Determine the General Condition for Vector Independence

A set of vectors is independent if the only linear combination that equals the zero vector is the trivial one, where all coefficients are zero. In other words, for vectors \( \{ v_1, v_2, \ldots, v_n \} \), we need \( c_1 v_1 + c_2 v_2 + \ldots + c_n v_n = 0 \) to imply \( c_1 = c_2 = \ldots = c_n = 0 \).

Check Independence for Set (a)

For the vectors \( \{1+x, 1-x, x+x^2\} \) in \( \mathbf{P}_2 \), write the equation \( c_1(1 + x) + c_2(1 - x) + c_3(x + x^2) = 0 \). This expands to:\[ c_1 + c_1x + c_2 - c_2x + c_3x + c_3x^2 = 0 \] Combine like terms: \( (c_1 + c_2) + (c_1 - c_2 + c_3)x + c_3x^2 = 0 \). This implies:\[ \begin{align*} c_1 + c_2 &= 0, \ c_1 - c_2 + c_3 &= 0, \ c_3 &= 0. \end{align*} \] Solving this system leads to \( c_1 = c_2 = c_3 = 0 \), proving independence.

Check Independence for Set (b)

For vectors \( \{x^2, x+1, 1-x-x^2\} \), write \( c_1x^2 + c_2(x + 1) + c_3(1 - x - x^2) = 0 \). This results in:\[ c_1x^2 + c_2x + c_2 + c_3 - c_3x - c_3x^2 = 0 \] Combine terms: \( (c_1 - c_3)x^2 + (c_2 - c_3)x + (c_2 + c_3) = 0 \). This implies:\[ \begin{align*} c_1 - c_3 &= 0, \ c_2 - c_3 &= 0, \ c_2 + c_3 &= 0. \end{align*} \] Solving these yields \( c_1 = c_2 = c_3 = 0 \), confirming independence.

Check Independence for Set of Matrices

For matrices \( \left\{ \begin{bmatrix} 1 & 1 \ 0 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \ 1 & -1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \ 0 & 1 \end{bmatrix} \right\} \), write the linear combination \( c_1 \begin{bmatrix} 1 & 1 \ 0 & 0 \end{bmatrix} + c_2 \begin{bmatrix} 1 & 0 \ 1 & 0 \end{bmatrix} + c_3 \begin{bmatrix} 0 & 0 \ 1 & -1 \end{bmatrix} + c_4 \begin{bmatrix} 0 & 1 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} \). This gives:\[ \begin{bmatrix} c_1 + c_2 & c_1 + c_4 \ c_2 + c_3 & -c_3 + c_4 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} \] which implies:\[ \begin{align*} c_1 + c_2 &= 0, \ c_1 + c_4 &= 0, \ c_2 + c_3 &= 0, \ -c_3 + c_4 &= 0. \end{align*} \] Solving gives \( c_1 = c_2 = c_3 = c_4 = 0 \), so the set is independent.

Key concepts

Vector Spaces

In mathematics, a vector space is a fundamental concept within linear algebra. It consists of a set of vectors, where you can perform vector addition and scalar multiplication, satisfying specific rules. Imagine vectors as arrows pointing in space, where each has a magnitude and direction. You can stretch or shrink these arrows or combine them using the operations defined above.
In more formal terms, a vector space over a field (like the real numbers) allows for the combination of vectors with scalars from that field. This combination follows certain axioms, such as closure, associativity, distributivity, and the existence of an additive identity (the zero vector) and inverses.
Understanding vector spaces is crucial because they provide a framework to study different mathematical objects, such as polynomials and matrices. Vector spaces are not limited to arrows; they can be function spaces or polynomial spaces, allowing them to be used in various applications, from physics to computer graphics.

• Vector Addition: If \( u \) and \( v \) are vectors, their sum \( u + v \) is also a vector.
• Scalar Multiplication: If \( c \) is a scalar, then\( c \cdot v \) is a vector.

Polynomials

Polynomials are expressions composed of variables and coefficients, using the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, in the vector space of polynomials with a degree less than or equal to two, denoted as \( \mathbf{P}_2 \), vectors are polynomials like \( 1 + x \), \( 1 - x \), and \( x + x^2 \).
These polynomial expressions can be part of a vector space because they can be added together and multiplied by scalars, fulfilling the vector space axioms. Moreover, just like regular vectors, polynomial vectors can be linearly independent. Linear independence for polynomials means that no polynomial in the set can be written as a linear combination of the others. This concept helps when dealing with polynomial regression and solving differential equations.
To test for independence, we set up an equation like \( c_1 p_1 + c_2 p_2 + ... + c_n p_n = 0 \) with the polynomials \( p_i \). A trivial solution, where all \( c_i = 0 \), implies the polynomials are independent.

Matrix Algebra

Matrix algebra involves the study of matrices, which are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. The field of matrix algebra covers various matrix operations, including addition, multiplication, and finding determinants and inverses. These operations are foundational for solving linear equations and transforming vector spaces.
In terms of vector spaces, matrices can be considered as vectors as well—in this case, vectors of matrices. For instance, matrices in \( \mathbf{M}_{22} \) have dimensions of 2x2 and can be part of vector spaces.
When discussing linear independence in matrix contexts, we are looking at whether a set of matrices can be combined linearly to produce the zero matrix, much like with traditional vectors. Suppose we have vectors composed of matrices instead of single numbers. In that case, we seek to find out if the only solution is the trivial one. Solving systems of linear equations using matrices, such as through Gaussian elimination, relies heavily on matrix algebra principles. Mathematical software often employs these techniques in computational calculations for engineering, physics, and computer science applications.