Word Count : 319

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We are given the set of vectors (polynomials) in $P_3$, the vector space of all polynomials of degree ≤ 3: $$ \{1 + x,\ x + x^2,\ 1 + x^3\} $$ We want to check whether these vectors are linearly independent. --- ### Step 1: Express each polynomial as a vector of coefficients Let us write each polynomial in the form $a_0 + a_1x + a_2x^2 + a_3x^3$. So, we convert the polynomials into 4-dimensional vectors (because $P_3$ has _____ ________ ________ ____ ____ ___ ___ _______ ___ _______ ________.
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