3. Let V be the vector space of polynomials with coefficients from R. a) Show that the following polynomials are linearly dependent: P1 (x) = x° – 3x + 1, P2(x) = 2x + 1, p3(x) = -x³ + 5x b) Show that...

3. Let V be the vector space of polynomials with coefficients from R.<br>a) Show that the following polynomials are linearly dependent:<br>P1 (x) = x° – 3x + 1,<br>P2(x) = 2x + 1, p3(x) = -x³ + 5x<br>b) Show that the set B below is a basis for V:<br>B = {pk(x) | k = 0, 1, ...} where pr:(x) =<br>Σ<br>j=0<br>so that po(x) = 1, p1(x) = 1+x, p2(x) = 1+ x + x², etc.<br>

Extracted text: 3. Let V be the vector space of polynomials with coefficients from R. a) Show that the following polynomials are linearly dependent: P1 (x) = x° – 3x + 1, P2(x) = 2x + 1, p3(x) = -x³ + 5x b) Show that the set B below is a basis for V: B = {pk(x) | k = 0, 1, ...} where pr:(x) = Σ j=0 so that po(x) = 1, p1(x) = 1+x, p2(x) = 1+ x + x², etc.

Jun 03, 2022

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3. Let V be the vector space of polynomials with coefficients from R. a) Show that the following polynomials are linearly dependent: P1 (x) = x° – 3x + 1, P2(x) = 2x + 1, p3(x) = -x³ + 5x b) Show that...

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