Question

Which of the following subsets of \(V\) are independent? a. \(V=\mathbf{P}_{2} ;\left\\{x^{2}+1, x+1, x\right\\}\) b. \(V=\mathbf{P}_{2} ;\left\\{x^{2}-x+3,2 x^{2}+x+5, x^{2}+5 x+1\right\\}\) c. \(V=\mathbf{M}_{22} ;\left\\{\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right],\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\right\\}\) d. \(V=\mathbf{M}_{22}\) \(\left\\{\left[\begin{array}{rr}-1 & 0 \\ 0 & -1\end{array}\right],\left[\begin{array}{rr}1 & -1 \\ -1 & 1\end{array}\right],\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right],\left[\begin{array}{rr}0 & -1 \\ -1 & 0\end{array}\right]\right\\}\) e. \(V=\mathbf{F}[1,2] ;\left\\{\frac{1}{x}, \frac{1}{x^{2}}, \frac{1}{x^{3}}\right\\}\) f. \(V=\mathbf{F}[0,1] ;\left\\{\frac{1}{x^{2}+x-6}, \frac{1}{x^{2}-5 x+6}, \frac{1}{x^{2}-9}\right\\}\)

Short answer

Sets (c) and (e) are independent.

Step-by-step solution

Understand vector space and independence

To determine if a set of vectors (subsets) is independent, we need to check if the only solution to a linear combination equaling the zero vector is when all coefficients are zero. For a space \( V \), a basis consists of independent vectors that span the space.

Check independence for Polynomials in \( \textbf{P}_2 \)

For sets (a) and (b), check if the polynomials are independent. (a) Set up the equation: \( a(x^2 + 1) + b(x + 1) + cx = 0 \). Solving gives: \,No unique solution for \( a, b, c \) implies dependence.(b) Set up the equation: \( a(x^2 - x + 3) + b(2x^2 + x + 5) + c(x^2 + 5x + 1) = 0 \). Solve to explore if \( a = b = c = 0 \) is the sole solution. It is not, indicating dependence.

Check independence for Matrices in \( \mathbf{M}_{22} \)

(c) Use the matrices to form an equation like \( a \begin{bmatrix}1 & 1 \ 0 &1\end{bmatrix} + b \begin{bmatrix}1 & 0 \ 1 &1\end{bmatrix} + c \begin{bmatrix}1 & 0 \ 0 &1\end{bmatrix} = \mathbf{0} \). Solve to see if \( a=b=c=0 \) is the only solution. It is, indicating independence.(d) Repeat similar steps for the matrices given. Form the linear combination and solve to verify independence, which it is not, as dependent solutions exist.

Check independence for Functions in specific intervals

(e) The functions are powers of \( \frac{1}{x} \). Setting up \( a\frac{1}{x} + b\frac{1}{x^2} + c\frac{1}{x^3} = 0 \) leads to the identities in terms of \( x \). Solving this shows independence since \( a = b = c = 0 \) is the sole solution.(f) The function forms involve rational expressions. Solving \( a(x^2 + x - 6)^{-1} + b(x^2 - 5x + 6)^{-1} + c(x^2 - 9)^{-1} = 0 \) yields non-unique solutions for \( a, b, c \), indicating dependence.

Key concepts

Polynomial Independence

Polynomial independence is a fundamental concept in linear algebra involving polynomial functions. When we talk about a polynomial being independent, we are essentially checking if there is a unique solution to a set of equations structured from these polynomials.
Let's say we have three polynomials: \( p(x) = x^2 + 1 \), \( q(x) = x + 1 \), and \( r(x) = x \). To determine if these are independent in the vector space \( \textbf{P}_2 \), we create a linear combination such that:

• \( a(p(x)) + b(q(x)) + c(r(x)) = 0 \)

This results in an equation that we solve to check if \( a = b = c = 0 \) is the only solution possible.
If it is, the polynomials are independent. If not, they are dependent. In the original exercise, solving similar combinations proved whether the given polynomials were independent or dependent.

Matrix Vector Space

A matrix vector space consists of all possible matrices of a given size, with operations of matrix addition and scalar multiplication. Matrices in this space can be checked for independence by forming equations from the matrices.
For example, consider matrices of size 2x2. To verify independence, you may encounter a set of matrices like the following:

• \( A = \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} \)
• \( B = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix} \)

Forming a linear combination looks like this:

• \( aA + bB + cC = \mathbf{0} \),

where \( A, B, C \) are matrices. Solve to find if \( a = b = c = 0 \) is the only solution. If it is, the matrices are independent.
In the exercise, some matrices were shown to be independent because they met this criterion, while others were dependent.

Function Independence

Function independence in the context of specific intervals involves determining if functions within a given set are linearly independent.
To assess this, you examine the structure of each function to confirm that no function in the set can be expressed as a combination of others, aside from a trivial solution.
For example, consider the set of functions: \( f_1(x) = \frac{1}{x} \), \( f_2(x) = \frac{1}{x^2} \), and \( f_3(x) = \frac{1}{x^3} \). We create the equation:

• \( a(f_1(x)) + b(f_2(x)) + c(f_3(x)) = 0 \)

We check if solving this equation results in \( a = b = c = 0 \) as the only solution, proving independence.
In the case explored by the exercise, certain functional sets were independent, satisfying the criterion, while others displayed dependence.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It is a foundational tool in both pure and applied mathematics.
Central to the study of linear algebra is the investigation of vector spaces and linear transformations. This includes work with polynomials, matrices, and functions.
Key concepts include:

• Vector space: a collection of vectors that can be added together and multiplied by scalars, maintaining closure under these operations.
• Linear independence: a set of vectors is independent if the only solution to their linear combination being the zero vector is all coefficients being zero.

In the given exercise, these ideas help determine whether polynomials, matrices, and functions are independent or dependent, showcasing linear algebra's utility in solving complex problems.