Question

In each case, find a basis for \(V\) among the given vectors. $$\begin{array}{l}\text { a. } V=\mathbb{R}^{3}, \\\\\quad\\{(1,1,-1),(2,0,1),(-1,1,-2),(1,2,1)\\}\end{array}$$ b. \(V=\mathbf{P}_{2},\left\\{x^{2}+3, x+2, x^{2}-2 x-1, x^{2}+x\right\\}\)

Short answer

a. \((1,1,-1), (2,0,1), (1,2,1)\) form a basis for \(\mathbb{R}^3\); b. \(x^2+3, x+2, x^2+x\) form a basis for \(\mathbf{P}_2\).

Step-by-step solution

Determine Independence

For part a, we begin by listing the given vectors in matrix form:\[A = \begin{bmatrix} 1 & 2 & -1 & 1 \ 1 & 0 & 1 & 2 \ -1 & 1 & -2 & 1 \end{bmatrix}\]To determine if they form a basis, we need to determine if the vectors are linearly independent by row reducing the matrix to echelon form.

Row Reduction

Perform row operations to bring the matrix to row-echelon form. After row reducing the matrix, you get:\[B = \begin{bmatrix} 1 & 0 & 3 & 0 \ 0 & 1 & -1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}\]Since the matrix has three leading ones, the original vectors have a rank of 3, indicating that they span \(\mathbb{R}^{3}\). Therefore, any three linearly independent vectors from these will form a basis.

Select Basis Vectors

From the row-reduced form, choose the columns corresponding to the pivot positions in the original matrix, obtaining the vectors \((1,1,-1), (2,0,1), \text{ and } (1,2,1)\) as a basis for \(\mathbb{R}^{3}\).

Determine Independence for Polynomials

For part b, consider the polynomials \(x^2+3, x+2, x^2-2x-1,\) and \(x^2+x\). Write them in the form of a coefficient matrix:\[C = \begin{bmatrix} 1 & 0 & 1 & 1 \ 0 & 1 & -2 & 1 \ 3 & 2 & -1 & 0 \end{bmatrix}\]Similarly, perform row reduction on this matrix.

Row Reduction for Polynomials

Continue row-reducing the matrix from step 4. The row-reduced form turns out to be:\[D = \begin{bmatrix} 1 & 0 & 1 & 0 \ 0 & 1 & -2 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}\]The non-zero rows of this row-reduced form correspond to the independent polynomials, showing that the set has a rank of 3.

Select Basis Polynomials

From the row-reduced form, identify the leading ones. The basis for \(\mathbf{P}_2\) consists of the polynomials \(x^2+3, x+2, x^2+x\) representing the independent columns of the original matrix.

Key concepts

Vector Spaces

A vector space is a collection of objects called vectors. In this context, vectors can be added together and multiplied by scalars to produce another vector in the same space. The vectors in a vector space can have different forms, such as ordered lists of numbers (tuples) or polynomials. For example, the vector space a. \(\mathbb{R}^{3}\) is made up of all possible ordered triples of real numbers (like \((1,1,-1)\)), while b. \(\mathbf{P}_{2}\) is the set of all polynomials of degree 2 or less, such as \(x^2+3\). Vector spaces have some key properties:

• Vectors can be added together.
• Vectors can be scaled by real numbers (scalars).
• There exist special vectors like the zero vector, which leaves other vectors unchanged when added to them.
• Adding vectors and scaling them is associative and commutative.

Understanding vector spaces is crucial for discussing ideas like bases and dimensions, which describe the structure of these spaces and how we can represent them efficiently.

Basis and Dimension

The basis of a vector space is a set of vectors that are both linearly independent and span the entire space. In simpler terms, these vectors can combine in different ways to build up every vector in the space. For instance, in the example of a. \(\mathbb{R}^{3}\), a basis could be the vectors \((1,1,-1)\), \((2,0,1)\), and \((1,2,1)\). These vectors can generate any vector in \(\mathbb{R}^{3}\) through suitable linear combinations.The dimension of a vector space is simply the number of vectors in its basis. For \(\mathbb{R}^{3}\), the dimension is 3 because we can express it via three linearly independent vectors.In the example of polynomials b. \(\mathbf{P}_{2}\), a basis could include polynomials like \(x^2+3\), \(x+2\), and \(x^2+x\). These are sufficient to describe any quadratic polynomial.

Linear Independence

Linear independence is a condition where no vector in a set can be represented as a linear combination of the others. When determining a basis, ensuring linear independence among vectors is crucial since it guarantees that each vector adds new dimensions to your space.Consider the vectors provided in example a. The original set \((1,1,-1),(2,0,1),(-1,1,-2),(1,2,1)\) is not fully linearly independent. By row reducing, we identified three vectors that maintain linear independence and span \(\mathbb{R}^3\).For b. In polynomials, the task is similar. We ensure the chosen polynomials do not express one another through linear combinations of degree 2 or fewer. Linear independence among chosen vectors or polynomials is key to forming an efficient basis for the vector space.

Row Reduction

Row reduction is a methodical way of simplifying matrices to determine linear independence and solve linear equations. It's an algebraic process that involves performing row operations to transform a matrix into row-echelon form.For the exercise's matrix, a. \ \[A = \begin{bmatrix} 1 & 2 & -1 & 1 \ 1 & 0 & 1 & 2 \ -1 & 1 & -2 & 1 \end{bmatrix}\] Row reducing this matrix helps us find pivot positions, which correspond to linearly independent vectors in the original set. After row reduction:\[B = \begin{bmatrix} 1 & 0 & 3 & 0 \ 0 & 1 & -1 & 0 \ 0 & 0 & 0 & 1 \end{bmatrix}\] indicates the linear independence of three vectors.In part b. For polynomials, row reducing enables us to identify independent sets even in non-numeric matrices, confirming which basis polynomials from \(\mathbf{P}_{2}\) are linearly independent. This approach simplifies complex sets, making solutions easier to understand and work with.