Question

In each case, find a basis of \(V\) containing \(\mathbf{v}\) and \(\mathbf{w}\) a. \(V=\mathbb{R}^{4}, \mathbf{v}=(1,-1,1,-1), \mathbf{w}=(0,1,0,1)\) b. \(V=\mathbb{R}^{4}, \mathbf{v}=(0,0,1,1), \mathbf{w}=(1,1,1,1)\) c. \(V=\mathbf{M}_{22}, \mathbf{v}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right], \mathbf{w}=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\) d. \(V=\mathbf{P}_{3}, \mathbf{v}=x^{2}+1, \mathbf{w}=x^{2}+x\)

Short answer

a) {(1,-1,1,-1), (0,1,0,1), (1,0,0,0), (0,0,1,0)}; b) {(0,0,1,1), (1,1,1,1), (1,0,0,0), (0,1,0,0)}; c) {[[1, 0], [0, 1]], [[0, 1], [1, 0]], [[1, 1], [0, 0]], [[0, 0], [1, 1]]}; d) {x^2+1, x^2+x, x, x^3}.

Step-by-step solution

Analyze Vectors and Subset in Part a

First, verify if the vectors \( \mathbf{v} = (1,-1,1,-1) \) and \( \mathbf{w} = (0,1,0,1) \) are linearly independent. If the vectors are independent, they form part of the basis. Write these vectors in matrix form as rows and perform row reduction to check for independence.

Row Reduction for Part a

Form a matrix with \( \begin{bmatrix} 1 & -1 & 1 & -1 \ 0 & 1 & 0 & 1 \ \end{bmatrix} \). Using elementary row operations, reduce this to row-echelon form. The resulting matrix shows both rows are nonzero and thus independent: \[\begin{bmatrix} 1 & -1 & 1 & -1 \ 0 & 1 & 0 & 1 \end{bmatrix}\] There's no zero row, indicating independence.

Determine Additional Vectors for a Basis in Part a

To complete the basis for \( \mathbb{R}^{4} \), add two more vectors. Choose two more standard basis vectors: \( \mathbf{e}_1 = (1,0,0,0) \) and \( \mathbf{e}_2 = (0,0,1,0) \), ensuring that adding them maintains linear independence.

Final Basis for Part a

The basis of \( V \) that includes \( \mathbf{v} \) and \( \mathbf{w} \) is: \[ \left\{ (1,-1,1,-1), (0,1,0,1), (1,0,0,0), (0,0,1,0) \right\} \].

Analyze Vectors and Subset in Part b

Examine vectors \( \mathbf{v} = (0,0,1,1) \) and \( \mathbf{w} = (1,1,1,1) \) for linear independence by arranging them in matrix form and reducing: \[ \begin{bmatrix} 0 & 0 & 1 & 1 \ 1 & 1 & 1 & 1 \end{bmatrix} \].

Row Reduction for Part b

Perform row operations to achieve: \[ \begin{bmatrix} 1 & 1 & 1 & 1 \ 0 & 0 & 1 & 1 \end{bmatrix} \]. The nonzero rows confirm \( \mathbf{v} \) and \( \mathbf{w} \) are independent.

Determine Additional Vectors for a Basis in Part b

To complete the basis in \( \mathbb{R}^{4} \), add two more vectors not in the span of \( \mathbf{v} \) and \( \mathbf{w} \). Examples are \( (1,0,0,0) \) and \( (0,1,0,0) \).

Final Basis for Part b

The basis in \( V \) that includes \( \mathbf{v} \) and \( \mathbf{w} \) is: \[ \left\{ (0,0,1,1), (1,1,1,1), (1,0,0,0), (0,1,0,0) \right\} \].

Analyze Matrices in Part c

For matrices, verify if \( \mathbf{v} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \) and \( \mathbf{w} = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} \) are independent by examining determinant properties or vectorizing them.

Vectorize and Manipulate in Part c

Treat these matrices as vectors: \( \mathbf{v'} = (1, 0, 0, 1) \) \( \mathbf{w'} = (0, 1, 1, 0) \). Convert them to a matrix and ensure row independence. After row reduction, all rows non-zero indicate independence.

Complete Basis in Part c

Extend the basis within \( \mathbf{M}_{22} \) by adding two matrices ensuring independence. Choose \( \begin{pmatrix} 1 & 1 \ 0 & 0 \end{pmatrix} \) and \( \begin{pmatrix} 0 & 0 \ 1 & 1 \end{pmatrix} \).

Final Basis for Part c

The basis containing \( \mathbf{v} \) and \( \mathbf{w} \) is: \[ \left\{ \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}, \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}, \begin{pmatrix} 1 & 1 \ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \ 1 & 1 \end{pmatrix} \right\} \].

Analyze Polynomials in Part d

Check if \( \mathbf{v} = x^{2} + 1 \) and \( \mathbf{w} = x^{2} + x \) are linearly independent by expressing them as a linear combination and solving for coefficients.

Independence Check for Part d

Write \( a(x^2 + 1) + b(x^2 + x) = 0 \), resulting in the equation \((a+b)x^2 + bx + a = 0\). Derive coefficients by equating each degree to zero, leading to \( a + b = 0, b = 0, a = 0 \), confirming independence.

Determine Missing Elements for Part d

To form a basis in \( \mathbf{P}_{3} \), add missing polynomials \( x \) and \( x^3 \) to ensure coverage of all degrees up to 3.

Final Basis for Part d

The basis containing \( \mathbf{v} \) and \( \mathbf{w} \) is: \[ \left\{ x^{2}+1, x^{2}+x, x, x^{3} \right\} \].

Key concepts

Linear Independence

In linear algebra, linear independence is a fundamental concept that helps us understand the relationship between vectors. Two or more vectors are said to be linearly independent if no vector among them can be written as a linear combination of the others.
This means that none of the vectors can be formed by adding or scaling the others.
This concept is crucial when you are trying to determine if a set of vectors can form a basis for a vector space.

• To check for linear independence, arrange the vectors as rows in a matrix and use row reduction.
• If you can reduce the matrix to a form where each vector becomes a pivot (leading 1), then the vectors are independent.
• If any rows become entirely zero, those vectors are dependent on the others.

Basis of Vector Spaces

The basis of a vector space is a set of vectors that are both linearly independent and span the entire space. A basis allows us to uniquely represent every element of a vector space as a linear combination of basis vectors.
It serves as the "building blocks" for the vector space.

• For an n-dimensional vector space, a basis must have exactly n vectors.
• Each vector in the space can be expressed as a combination of these basis vectors.
• To find a basis, ensure the vectors you choose are linearly independent and collect enough such vectors to span the space.

In practical terms, finding a basis can often involve adding vectors to an independent set until it spans the space.

Row Reduction

Row reduction is a technique used with matrices to simplify them into a form called row-echelon form. It is mainly used to solve systems of linear equations, but it is also very useful in determining linear independence.
When you prepare your matrix with your vectors as rows, row reduction helps you see if those vectors are independent.

• Use elementary row operations like swapping rows, multiplying rows by scalars, and adding/subtracting scalar multiples of rows.
• The goal is to reach row-echelon form where leading entries are 1 (this signals independence).
• If the matrix reduces without any rows turning entirely into zeros, your vectors are independent.

Polynomials

Polynomials are expressions consisting of variables raised to non-negative integer powers and their coefficients. In vector spaces, polynomials can be treated similarly to numerical vectors.
Polynomials of degree n will form a vector space, denoted usually by \( \mathbf{P}_n \), which includes all polynomials of that degree or less.

• To check if a set of polynomials is independent, express one polynomial as a combination of others and solve—if only trivial solutions (e.g., all coefficients equal zero) exist, they are independent.
• In forming a basis for polynomial spaces, ensure polynomials cover all powers up to the space's degree.
• Like with numerical vectors, a set of independent polynomials can serve as a basis for polynomial spaces.

This understanding is helpful in solving polynomial equations and in analysis where spectral methods are relevant.