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Problem 176

# Let $$\mathrm{P}_{4}$$, denote the vector space of all polynomials of degree at most equal to four. Let $$\mathrm{V}$$ be the subspace of $$\mathrm{P}_{4}$$. spanned by $\mathrm{S}=\left\\{\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}\right\\}$$where$$\alpha_{1}=\mathrm{t}^{4}+\mathrm{t}^{2}+2 \mathrm{t}+1$ $$\alpha_{2}=t^{4}+t^{2}+2 t+2, \alpha_{3}=2 t^{4}+t^{3}+t+2$$ and $$\alpha_{4}=t^{4}+t^{3}-t^{2}$$ $$-\mathrm{t}$$. Find a basis for $$\mathrm{V}$$.

Expert verified
The basis for subspace V is the set B = { $$α_1, α_3, α_4$$ }, where $$α_1 = t^4 + 2t^3 + t^2 + 1$$, $$α_3 = -\frac{1}{3}(-3t^3 - 2t^2 + t)$$, and $$\alpha_4 = -t + 1$$.
See the step by step solution

## Step 1: Set up the matrix A from S

Construct a matrix A using the coefficients of the given polynomials in the set S as row vectors. A = $\begin{pmatrix} 1 & 2 & 1 & 0 & 1\\ 1 & 2 & 1 & 0 & 1\\ 2 & 1 & 0 & 1 & 2\\ 1 & -1 & -1 & 1 & 0 \end{pmatrix}$

## Step 2: Perform Gaussian elimination on matrix A

To find linearly independent polynomials, we need to perform Gaussian elimination (row reduction) on the matrix A. Row 2 can be eliminated, since it is the same as Row 1. Let's also eliminate Row 4: R4 - R1 -> R4 This results in the following matrix: A = $\begin{pmatrix} 1 & 2 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & 0\\ 2 & 1 & 0 & 1 & 2\\ 0 & -3 & -2 & 1 & -1 \end{pmatrix}$ Now, eliminate Row 3: R3 - 2*R1 -> R3 This leads to the following matrix: A = $\begin{pmatrix} 1& 2 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & 0\\ 0 & -3 & -2 & 1 & 0\\ 0 & -3 & -2 & 1 & -1 \end{pmatrix}$ Finally, we can eliminate Row 4: R4 - R3 -> R4 This results in the following matrix: A = $\begin{pmatrix} 1 & 2 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & 0\\ 0 & -3 & -2 & 1 & 0\\ 0 & 0 & 0 & 0 & 1 \end{pmatrix}$

## Step 3: Determine the basis for subspace V

Now that our matrix A is in row echelon form, we can determine the basis for subspace V. The non-zero rows of the matrix A correspond to linearly independent polynomials in the set S. According to the row reduced matrix, the linearly independent polynomials are: $$α_1 = t^4 + 2t^3 + t^2 + 1$$ $$α_3 = -\frac{1}{3}(-3t^3 - 2t^2 + t)$$ $$\alpha_4 = -t + 1$$ So, the basis for subspace V is the set: B = { $$α_1, α_3, α_4$$ } This basis consists of three linearly independent polynomials that span the subspace V.

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